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href="../../../mathsanalysis.html"><i class="fa fa-home"></i></a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><a href="../537/algebra.html">Algebra</a><i class="fa fa-fw fa-chevron-right divider"></i></li><li><span class="gray">Indices and Logarithms</span></li> <span class="pull-right" style="color: #555" title="Suggested study time: 30 minutes"><i class="fa fa-clock-o"></i> 30'</span> </ol> <article id="main-article"> <p><img alt="" src="../../files/algebra/logarithms/main-1.jpg" style="float: left; width: 100px; height: 100px;"><img class="sibico" src="../../../img/sibico/hl-blue.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL"> <img class="sibico" src="../../../img/sibico/sl-blue.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL"> Logarithms are useful for solving problems involving indices (or exponents). In fact, logarithms are just indices in disguise! The definition of a logarithm helps you to see its equivalence with indices: <span class="math-tex">\(a^x=b \quad\Leftrightarrow \quad x=\log _{ a }{ b } \)</span> . Questions about logarithms come up in all different areas of the course so it is important to be comfortable using them. This page will help you with that!</p> <hr class="hidden-separator"> <div class="panel panel-turquoise panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Key Concepts</p> </div> </div> <div class="panel-body"> <div> <p>On this page, you should learn to</p> <ul> <li>Convert index equations into logarithms (and vice versa)</li> </ul> <p style="margin-left: 120px;"><span class="math-tex">\(a^x=b \quad\Leftrightarrow \quad x=\log _{ a }{ b } \)</span></p> <p style="margin-left: 120px;"><span class="math-tex">\(e^x=b \quad\Leftrightarrow \quad x=\ln b\)</span></p> <ul> <li>Use the laws of indices (exponents) and their equivalents for logarithms</li> </ul> <table border="0" cellpadding="0" cellspacing="0" style="width:100%;"> <tbody> <tr> <td style="text-align: right;">Laws of Indices </td> <td>Laws of Logarithms</td> </tr> <tr> <td style="text-align: right;"><span class="math-tex">\(m^x \times m^y=m^{x+y} \quad \Leftrightarrow \quad\)</span></td> <td><span class="math-tex">\(\log _{ c }{ ab }=\log _{ c }{ a } +\log _{ c }{ b } \)</span></td> </tr> <tr> <td style="text-align: right;"> </td> <td> </td> </tr> <tr> <td style="text-align: right;"><span class="math-tex">\(m^x \div m^y=m^{x-y} \quad \Leftrightarrow \quad\)</span></td> <td><span class="math-tex">\(\log _{ c }{ \frac{a}{b} } = \log _{ c }{ a } -\log _{ c }{ b }\)</span></td> </tr> <tr> <td style="text-align: right;"> </td> <td> </td> </tr> <tr> <td style="text-align: right;"><span class="math-tex">\((m^x)^y=m^{x \times y} \quad \Leftrightarrow \quad\)</span></td> <td><span class="math-tex">\(r\log _{ c }{ a }=\log _{ c }{ a ^r}\)</span></td> </tr> </tbody> </table> <ul> <li> Change the base of a logarithm</li> </ul> <p style="margin-left: 120px;"><span class="math-tex">\(\log _{ b }{ a } =\frac { \log _{ c }{ a } }{ \log _{ c }{ b } } \)</span></p> <ul> <li>Solve index equations</li> </ul> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Essentials</p> </div> </div> <div class="panel-body"> <p>The following videos will help you understand all the concepts from this page</p> <div class="panel panel-yellow panel-has-colored-body panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Understanding Logarithms</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="472"> <p>One of the important things about using logarithms is to be able to see them as indices, but just written in another way <span></span><span class="math-tex">\(a^x=b \quad\Leftrightarrow \quad x=\log _{ a }{ b } \)</span></p> <p><span>Once you understand the conversion, then these questions become more about understanding indices.</span></p> <p>In the following video, we will look at these 5 examples</p> <p style="margin-left: 40px;"><span class="math-tex">\(1. \quad \log _{ 2 }{ 32 }=x \\ 2. \quad \log _{ 10}{ x }= 3\\ 3. \quad\ln{ 1 }=x \\ 4. \quad\log _{ 5 }{ \frac{1}{125} }=x \\ 5. \quad\log _{ 8}{ \frac{1}{4} } =x\\\)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/281652939"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/converting-logs-to-indices.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/converting-logs-to-indices.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-expandable panel-has-border"> <div class="panel-heading"> <div> <p>Laws of Logarithms</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="473"> <p>There are laws of indices and these have equivalents for logarithms</p> <table border="0" cellpadding="0" cellspacing="0" style="width:100%;"> <tbody> <tr> <td style="text-align: right;">Laws of Indices</td> <td>Laws of Logarithms</td> </tr> <tr> <td style="text-align: right;"><span class="math-tex">\(m^x \times m^y=m^{x+y} \quad \Leftrightarrow \quad\)</span></td> <td><span class="math-tex">\(\log _{ c }{ ab }=\log _{ c }{ a } +\log _{ c }{ b } \)</span></td> </tr> <tr> <td style="text-align: right;"> </td> <td> </td> </tr> <tr> <td style="text-align: right;"><span class="math-tex">\(m^x \div m^y=m^{x-y} \quad \Leftrightarrow \quad\)</span></td> <td><span class="math-tex">\(\log _{ c }{ \frac{a}{b} } = \log _{ c }{ a } -\log _{ c }{ b }\)</span></td> </tr> <tr> <td style="text-align: right;"> </td> <td> </td> </tr> <tr> <td style="text-align: right;"><span class="math-tex">\((m^x)^y=m^{x \times y} \quad \Leftrightarrow \quad\)</span></td> <td><span class="math-tex">\(r\log _{ c }{ a }=\log _{ c }{ a ^r}\)</span></td> </tr> </tbody> </table> <p><span></span><span>These are useful for solving many different types of problem.</span></p> <p>In the following video, we will look at some examples that require us to use these laws:</p> <ol> <li>Show that the equation <span class="math-tex">\(\log _{ 2 }{ (x+2) } +\log _{ 2 }{ (x+3) } =1\)</span> has only one solution and state its value.</li> <li>Make <strong><em>x </em></strong>the subject of the following formula</li> </ol> <p style="margin-left: 120px;"><span class="math-tex">\(2+\log _{ 5 }{ x=3\log _{ 5 }{ y } } \)</span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/281755492"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/logarithm-laws.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/logarithm-laws.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-expandable panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Solving Index Equations</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="470"> <p>In the following video we look at solving equations involving indices. We will solve the following equations:</p> <ol> <li style="margin-left: 80px;"><span class="math-tex">\(2^x=12\)</span></li> <li style="margin-left: 80px;"><span class="math-tex">\(3^{x+1}=\frac{1}{27}\)</span></li> <li style="margin-left: 80px;"><span class="math-tex">\(5^x=2^{x+1}\)</span></li> </ol> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/281594546"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/solving-index-equations.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/solving-index-equations.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-yellow panel-has-colored-body panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Solving Logarithm Equations</p> </div> </div> <div class="panel-body"> <div class="smart-object center" data-id="471"> <p>Here is a good example of an exam-style question. We are solving an equation that involves both logarithmic and trigonometric functions.</p> <p>Solve <span class="math-tex">\(\log _{ 4 }{ \frac { cosx }{ 3 } } +\log _{ 4 }{ cosx=-1 } \)</span> , for <span class="math-tex">\(-\pi<x<\pi\)< span=""></x<\pi\)<></span></p> <div class="video-embed vimeo"><iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" mozallowfullscreen="" webkitallowfullscreen="" height="420" width="100%" src="https://player.vimeo.com/video/281625782"></iframe></div> <h4><span></span><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span><span></span> Notes from the video</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/solving-log-equation-with-trig.pdf" target="_blank">here</a></p> <p style="text-align: center;"><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/solving-log-equation-with-trig.pdf" width="640"></iframe></p> </section> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-violet"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Summary</p> </div> </div> <div class="panel-body"> <div> <p><iframe align="middle" frameborder="1" height="480" scrolling="yes" src="../../files/algebra/logarithms/revision-notes_exponents_and_logarithms.pdf" width="640"></iframe></p> <p>Print from <a href="../../files/algebra/logarithms/revision-notes_exponents_and_logarithms.pdf" target="_blank">here</a></p> </div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-green"> <div class="panel-heading"><a class="expander pull-right" href="#"><span class="fa fa-plus"></span></a> <div> <p>Test Yourself</p> </div> </div> <div class="panel-body"> <p>Here is a quiz that will test your basic understanding of logarithms</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#ed589b24"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="ed589b24"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Logarithms and indices equivalence <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-607-543" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log_28=\)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large 2^3=8\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log_381=\)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large 3^4=81\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log_{10}0.1=\)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-1"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large 10^{-1}=\frac{1}{10^1}=0.1\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log_42=\)</span></p><p style="text-align: right;">Give your answer as a decimal</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large 4^{0.5}=\sqrt{4}=2\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log_{100}\sqrt{100}=\)</span></p><p style="text-align: right;">Give your answer as a decimal</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large 100^{0.5}=\sqrt{100}\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log_a\frac{1}{\sqrt{a}}=\)</span></p><p style="text-align: right;">Give your answer as a decimal</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large a^{-0.5}=\frac{1}{a^{\frac{1}{2}}}=\frac{1}{\sqrt{a}}\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log_aa^3=\)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log_48=\)</span></p><p style="text-align: right;">Give your answer as a decimal</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large 4^{\frac{3}{2}}=(\sqrt{4})^3=2^3=8\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log1=\)</span></p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large 10^0=1\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out</p><p><span class="math-tex">\(\large\log_4\frac{1}{2}=\)</span></p><p style="text-align: right;">Give your answer as a decimal</p></div><div class="q-answer"><p> <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>From the definition of logs, we know that</p><p><span class="math-tex">\(\large\log_ab=x \ \Leftrightarrow \ a^x=b\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large 4^{-\frac{1}{2}}=\frac{1}{4^{\frac{1}{2}}}=\frac{1}{\sqrt{4}}=\frac{1}{2}\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here is a quiz about the laws of logarithms</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#36d8b4e1"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="36d8b4e1"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Log Laws <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-609-543" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large\log2+\log5=\log a\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_axy=\log_ax+\log_ay\)</span></p><hr class="hidden-separator">In this case, <span class="math-tex">\(\large\log2+\log5=\log (2\times5)\)</span></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large\log3+\log4=\log a\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="12"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_axy=\log_ax+\log_ay\)</span></p><hr class="hidden-separator"><p>In this case, <span class="math-tex">\(\large\log3+\log4=\log (3\times4)\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large\log15-\log3=\log a\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_a\frac{x}{y}=\log_ax-\log_ay\)</span></p><hr class="hidden-separator"><p>In this case, <span class="math-tex">\(\large\log15-\log3=\log (\frac{15}{3})=\log5\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large\log20-\log2-\log5=\log a\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_a\frac{x}{y}=\log_ax-\log_ay\)</span></p><hr class="hidden-separator"><p>In this case, <span class="math-tex">\(\large\log20-\log2-\log5=\log (20\div2)-\log5=\log(10\div5)=\log2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large2\log3=\log a\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_ax^m=m\log_ax\)</span></p><hr class="hidden-separator"><p>In this case, <span class="math-tex">\(\large2\log3=\log 3^2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large3\log2=\log a\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_ax^m=m\log_ax\)</span></p><hr class="hidden-separator"><p>In this case, <span class="math-tex">\(\large3\log2=\log 2^3\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large a\log3=\log 81\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_ax^m=m\log_ax\)</span></p><hr class="hidden-separator"><p>In this case, <span class="math-tex">\(\large \log 81=\log 3^4=4\log3\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large2\log4-0.5\log16=0.5\log a\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="144"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_a\frac{x}{y}=\log_ax-\log_ay\)</span></p><p><span class="math-tex">\(\large\log_ax^m=m\log_ax\)</span></p><hr class="hidden-separator"><p>In this case,</p><p><span class="math-tex">\(\large2\log4-0.5\log16=\log4^2-\log\sqrt{16}=\log16-\log4=\log12\)</span></p><p><span class="math-tex">\(\large\log12=0.5\log144\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve for x</p><p><span class="math-tex">\(\large\log_2(2x+4)-\log_2(x-1)=3\)</span></p></div><div class="q-answer"><p><strong><em>x </em></strong>= <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_a\frac{x}{y}=\log_ax-\log_ay\)</span></p><hr class="hidden-separator"><p>In this case,</p><p><span class="math-tex">\(\large\log_2(2x+4)-\log_2(x-1)=\log_2\frac{2x+4}{x-1}\)</span></p><p>Therefore,</p><p><span class="math-tex">\(\large\log_2\frac{2x+4}{x-1}=3\)</span></p><p><span class="math-tex">\(\large\frac{2x+4}{x-1}=2^3\\
\large\frac{2x+4}{x-1}=8\\\)</span></p><p>Solve the equation</p><p><span class="math-tex">\(\large 2x+4=8(x-1)\\
2x+4=8x-8\\
12=6x\\
x=2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve for x</p><p><span class="math-tex">\(\large\log4x-\log(x-3)=1\)</span></p></div><div class="q-answer"><p><strong><em>x </em></strong>= <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_a\frac{x}{y}=\log_ax-\log_ay\)</span></p><hr class="hidden-separator"><p>In this case,</p><p><span class="math-tex">\(\large\log4x-\log(x-3)=\log\frac{4x}{x-3}\)</span></p><p>We have <span class="math-tex">\(\large\log\frac{4x}{x-3}=1\)</span></p><p>Remember that <span class="math-tex">\(\large \log\)</span> means <span class="math-tex">\(\large \log_{10}\)</span></p><p>Therefore,</p><p><span class="math-tex">\(\large\frac{4x}{x-3}=10^1\)</span></p><p>Solve the equation</p><p><span class="math-tex">\(\large 4x=10(x-3)\\
4x=10x-30\\
30=6x\\
x=5\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here is a quiz that tests your knowledge of changing bases of logarithms</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#b914261c"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="b914261c"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Logarithms - change of base <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-610-543" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large \log_27=\frac{\log a}{\log b}\)</span></p><p>work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="7"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large \log3=\frac{\log_2a}{\log_2b} \)</span></p><p>work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p><p>Remember that <span class="math-tex">\(\large \log\)</span> means <span class="math-tex">\(\large \log_{10}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large \log_35=\frac{\log_6a}{\log_b3}\)</span></p><p>work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p><p>In this case,</p><p><span class="math-tex">\(\large \log_35=\frac{\log_65}{\log_63}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large \log_ab=\frac{\log 8}{\log 4}\)</span></p><p>work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large \log_a5=\frac{\log_25}{3}\)</span></p><p>work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p><hr class="hidden-separator"><p>We use this</p><p><span class="math-tex">\(\large 3=\log_28\)</span></p><p>and so</p><p><span class="math-tex">\(\large \log_85=\frac{\log_25}{\log_28}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <strong><em>a</em></strong> and <strong><em>b</em></strong> if</p><p><span class="math-tex">\(\large -\log_ab=\frac{\log \frac{1}{5}}{\log 3}\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p><hr class="hidden-separator"><span class="math-tex">\(\large -\log_ab=\log_a{b^{-1}}=\log_a{\frac{1}{b}}=\frac{\log \frac{1}{b}}{\log a}\)</span><p>Therefore,</p><p><span class="math-tex">\(\large \frac{\log \frac{1}{b}}{\log a}=
\frac{\log \frac{1}{5}}{\log 3}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Work out <strong><em>a</em></strong> and <strong><em>b </em></strong>if</p><p><span class="math-tex">\(\large \frac{\log 0.25}{\log 8}=-\log_ab\)</span></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p><hr class="hidden-separator"><p><span class="math-tex">\(\large \frac{\log 0.25}{\log 8}=\log_80.25\)</span></p><p>And,</p><p><span class="math-tex">\(-\log_80.25=\log_8(\frac{1}{4})^{-1}=\log_84\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large \log_{5^2}3^5=\frac{5\log 3}{a\log b}\)</span></p><p>work out <strong><em>a</em></strong> and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p><hr class="hidden-separator"><p><span class="math-tex">\(\large \log_{5^2}3^5=\frac{\log 3^5}{\log 5^2}=\frac{5\log 3}{2\log 5}\)</span></p><p> </p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>If <span class="math-tex">\(\large \log_2x=\log_x2\)</span></p><p>work out 2 possible values of <strong><em>x</em></strong></p><p style="text-align: right;">Put values in order of size</p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span> or <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p><hr class="hidden-separator"><p>In this case, we can write the right-hand side of the equation in terms of log base 2</p><p><span class="math-tex">\(\large \log_x2=\frac{\log_22}{\log_2x}=\frac{1}{\log_2x}\)</span></p><p>Our equation becomes</p><p><span class="math-tex">\(\large \log_2x=\frac{1}{\log_2x}\\
\large(\log_2x)^2=1\\
\log_2x=\pm1\)</span></p><p><span class="math-tex">\(\large x=2^1 \)</span> or <span class="math-tex">\(\large x=2^{-1} \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve for <strong><em>x</em></strong></p><p><span class="math-tex">\(\large \log_2x-1=\log_4x\)</span></p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>The change of base formula from the formula booklet is useful here</p><p><span class="math-tex">\(\large \log_ax=\frac{\log_bx}{\log_ba}\)</span></p><hr class="hidden-separator"><p>In this case, we can write the right-hand side of the equation in terms of log base 2</p><p><span class="math-tex">\(\large \log_4x=\frac{\log_2x}{\log_24}=\frac{\log_2x}{2}\)</span></p><p>Our equation becomes</p><p><span class="math-tex">\(\large \log_2x-1=\frac{\log_2x}{2}\\
\large 2(\log_2x-1)=\log_2x\\
\large 2\log_2x-2=\log_2x\\
\)</span></p><p>Rearrange the equation</p><p><span class="math-tex">\(\large 2\log_2x-\log_2x=2\\
\large \log_2x=2\\x=2^2\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here is a quiz about solving index equations. You will not need to use logarithms to solve all of these.</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#0bbabd4b"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="0bbabd4b"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Index Equations <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-611-543" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Solve for <strong><em>x</em></strong></p><p><strong><em><span class="math-tex">\(\large 2^x=8\)</span></em></strong></p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p><strong><em><span class="math-tex">\(\large 2^3=8\)</span></em></strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve for <strong><em>x</em></strong></p><p><strong><em><span class="math-tex">\(\large 4^{-x}=2\)</span></em></strong></p><p style="text-align: right;">Write your answer as a decimal</p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="-0.5"> <span class="review"></span></p></div><div class="q-explanation"><p>Write 4 as a power of 2</p><p><strong><em><span class="math-tex">\(\large 4^{-x}=(2^2)^{-x}\)</span></em></strong></p><p>The index equation becomes</p><p><strong><em><span class="math-tex">\(\large 2^{-2x}=2^1\)</span></em></strong></p><p>Therefore,</p><p>-2x = 1</p><p>x = -0.5</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve for <strong><em>x</em></strong></p><p><strong><em><span class="math-tex">\(\large 3^x=\frac{1}{27}\)</span></em></strong></p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="-3"> <span class="review"></span></p></div><div class="q-explanation"><p><strong><em><span class="math-tex">\(\large 3^x=\frac{1}{27}\\
\large 3^x=\frac{1}{3^3}\\
\large 3^x=3^{-3}\)</span></em></strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve for <strong><em>x</em></strong></p><p><strong><em><span class="math-tex">\(\large 16^x=64\)</span></em></strong></p><p style="text-align: right;">Write your answer as a decimal</p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></p></div><div class="q-explanation"><p>6 and 64 as powers of 4 (or 2)</p><p><strong><em><span class="math-tex">\(\large 16=4^2 \ , \ 64=4^3\)</span></em></strong></p><p>The equation becomes</p><p><strong><em><span class="math-tex">\(\large (4^2)^x=4^3\\
2x=3 \\
x=1.5\)</span></em></strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve for <strong><em>x</em></strong></p><p><strong><em><span class="math-tex">\(\large 5^{x+2}=\frac{1}{25}\)</span></em></strong></p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="-4"> <span class="review"></span></p></div><div class="q-explanation"><p>Write the right-hand side of the equation in terms of a power of 5</p><p><strong><em><span class="math-tex">\(\large 5^{x+2}=\frac{1}{5^2}\\
\large 5^{x+2}=5^{-2}\\
\)</span></em></strong></p><p>Now we can equate the powers</p><p>x+2 = -2<br>x = -4</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve for <strong><em>x</em></strong></p><p><strong><em><span class="math-tex">\(\large4^{x-2}=\frac{1}{2}\)</span></em></strong></p><p style="text-align: right;">Write your answer as a decimal</p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="1.5"> <span class="review"></span></p></div><div class="q-explanation"><p>Re-write the equation as powers of 2</p><p><strong><em><span class="math-tex">\(\large4^{x-2}=\frac{1}{2}\\
\large(2^2)^{x-2}=2^{-1}\\
\large2^{2x-4}=2^{-1}\)</span></em></strong></p><p>Now we can equate the powers</p><p>2x - 4 = -1<br>2x = 3<br>x = 1.5</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Solve for <strong><em>x</em></strong></p><p><strong><em><span class="math-tex">\(\large 4^{x+1}=8^{x-2}\)</span></em></strong></p></div><div class="q-answer"><p><strong><em>x</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p></div><div class="q-explanation"><p>Re-write the equations as powers of 2</p><p><strong><em><span class="math-tex">\(\large 4^{x+1}=8^{x-2}\\
\large (2^2)^{x+1}=(2^3)^{x-2}\\
\large 2^{2x+2}=2^{3x-6}\)</span></em></strong></p><p>Now we can equate the powers</p><p>2x + 2 = 3x - 6</p><p>8 = x</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The solution to</p><p><strong><em><span class="math-tex">\(\large 5^x=2^{x+1}\)</span></em></strong></p><p>can be written in the form <span class="math-tex">\(\large x=\frac{2}{(\log_2a-1)}\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong>= <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p></div><div class="q-explanation"><p>In order to solve this eqution, we have to take logs of both sides.</p><p>It is slightly easier if we take logs base 2</p><p><strong><em><span class="math-tex">\(\large \log_2(5^x)=\log_2(2^{x+1})\)</span></em></strong></p><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_ax^m=m\log_ax\)</span></p><p>Our equation becomes</p><p><strong><em><span class="math-tex">\(\large x\log_25=(x+2)\log_22\\
\large x\log_25=(x+2)\times1\\
\large x\log_25=x+2\\
\large x\log_25-x=2\\
\large x(\log_25-1)=2\\
\large x=\frac{2}{(\log_25-1)}\\\)</span></em></strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The solution to</p><p><strong><em><span class="math-tex">\(\large 3^{x+1}=8^{x-1}\)</span></em></strong></p><p>can be written in the form <span class="math-tex">\(\large x=\frac{1+\log_3a}{\log_3b-1}\)</span></p><p>Work out <strong><em>a </em></strong>and <strong><em>b</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong>= <input type="text" style="height: auto;" data-c="8"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="8"> <span class="review"></span> </p></div><div class="q-explanation"><p>In order to solve this eqution, we have to take logs of both sides.</p><p>It is slightly easier if we take logs base 3</p><p><strong><em><span class="math-tex">\(\large \log_3(3^{x+1})=\log_3(8^{x-1})\)</span></em></strong></p><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_ax^m=m\log_ax\)</span></p><p>Our equation becomes</p><p><strong><em><span class="math-tex">\(\large (x+1)\log_33=(x-1)\log_38\\
\large x+1=(x-1)\log_38\\
\large x+1=x\log_38-\log_38\\
\large 1+\log_38=x\log_38-x\\
\large 1+\log_38=x(\log_38-1)\\
\large x=\frac{1+\log_38}{\log_38-1}\)</span></em></strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The solution to</p><p><strong><em><span class="math-tex">\(\large 2\cdot3^x=4^{x-1}\)</span></em></strong></p><p>can be written in the form <span class="math-tex">\(\large x=\frac{a}{b-\log_23}\)</span></p><p>Work out <strong><em>a</em></strong></p></div><div class="q-answer"><p><strong><em>a</em></strong>= <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p></div><div class="q-explanation"><p>In order to solve this eqution, we have to take logs of both sides.</p><p>It is slightly easier if we take logs base 2</p><p><strong><em><span class="math-tex">\(\large \log_2(2\cdot3^x)=\log_2(4^{x-1})\\
\)</span></em></strong></p><p>From the fomula booklet, we know the following about logs</p><p><span class="math-tex">\(\large\log_ax^m=m\log_ax\)</span></p><p><span></span><span class="math-tex">\(\large\log_axy=\log_ax+\log_ay\)</span><span></span></p><p>Our equation becomes</p><p><strong><em><span class="math-tex">\(\large \log_22+ \log_2(3^x)=\log_2(4^{x-1})\\
\large 1+ x\log_23=(x-1)\log_24\\
\large 1+ x\log_23=(x-1)\times2\\
\large 1+ x\log_23=2x-2\\
\large 3= 2x-x\log_23\\
\large 3= x(2-\log_23)\\
\large x=\frac{3}{2-\log_23}
\)</span></em></strong></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> <hr class="hidden-separator"> <p>Here is a quiz that practises ALL the skills from this page</p> <br><a class="btn btn-primary btn-block text-center" data-toggle="modal" href="#1bfbe659"><i class="fa fa-play"></i> START QUIZ!</a><div class="modal fade modal-slide-quiz" id="1bfbe659"> <div class="modal-dialog" style="width: 95vw; max-width: 960px"> <div class="modal-content"> <div class="modal-header slide-quiz-title"> <h4 class="modal-title" style="width: 100%;"> Logarithms <strong class="q-number pull-right"> <span class="counter">1</span>/<span class="total">1</span> </strong> </h4> </div> <div class="modal-body p-xs-3"> <div class="slide-quiz" data-stats="11-205-543" style="opacity: 0"> <div class="exercise shadow-bottom"><div class="q-question"><p>Each of these logarithm equations has an equivalent index equation. Match them up!</p><p>Remember that logx means log<sub>10</sub>x</p></div><div class="q-answer"><table border="0" cellpadding="0" cellspacing="0" style="width:100%;"><tbody><tr><td style="text-align: right;"><p>10<sup>2</sup>=100 <=> <input type="text" style="height: auto;" data-c="log100 = 2"> <span class="review"></span></p></td><td><p> <span class="q-text-draggable draggable" draggable="true">log0.1 = -1</span></p></td></tr><tr><td style="text-align: right;"><p>10<sup>3</sup>=1000 <=> <input type="text" style="height: auto;" data-c="log1000 = 3"> <span class="review"></span></p></td><td><p> <span class="q-text-draggable draggable" draggable="true">log10 = 1</span></p></td></tr><tr><td style="text-align: right;"><p>10<sup>-1</sup>=0.1<=> <input type="text" style="height: auto;" data-c="log0.1 = -1"> <span class="review"></span></p></td><td><p> <span class="q-text-draggable draggable" draggable="true">log100 = 2</span></p></td></tr><tr><td style="text-align: right;"><p>10<sup>1</sup>=10 <=> <input type="text" style="height: auto;" data-c="log10 = 1"> <span class="review"></span></p></td><td><p> <span class="q-text-draggable draggable" draggable="true">log1000 = 3</span></p></td></tr><tr><td style="text-align: right;"><p>10<sup>0</sup>=1 <=> <input type="text" style="height: auto;" data-c="log1 = 0"> <span class="review"></span></p></td><td> <span class="q-text-draggable draggable" draggable="true">log1 = 0</span></td></tr></tbody></table></div><div class="q-explanation"><p>log<sub>10</sub>b=a <=> 10<sup>a</sup>=b</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Write the value of these logarithms</p></div><div class="q-answer"><ol><li>log<sub>2</sub>8 = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></li><li><span class="math-tex">\(\log _{ 5 }{ \frac { 1 }{ 25 } } \)</span> = <input type="text" style="height: auto;" data-c="-2"> <span class="review"></span></li><li>log<sub>4</sub>2 = <input type="text" style="height: auto;" data-c="0.5"> <span class="review"></span></li><li>lne = <input type="text" style="height: auto;" data-c="1"> <span class="review"></span></li><li>ln1 = <input type="text" style="height: auto;" data-c="0"> <span class="review"></span></li></ol></div><div class="q-explanation"><ol><li>2<sup>3</sup> = 8</li><li>5<sup>-2</sup> = <span class="math-tex">\(\frac{1}{5^2}=\frac{1}{25}\)</span></li><li>4<sup>0.5</sup> = <span class="math-tex">\(\sqrt{4}=2\)</span></li><li>lne = log<sub>e</sub>e=1 <=>e<sup>1</sup>=e</li><li>ln1 = log<sub>e</sub>1=0 <=>e<sup>0</sup>=1</li></ol></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Fill in the blanks below</p></div><div class="q-answer"><p>log2 + log3 = log <input type="text" style="height: auto;" data-c="6"> <span class="review"></span></p><p>log12 - log4 = log <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>log6 + log4 - log8 = log <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p>2log3 = log <input type="text" style="height: auto;" data-c="9"> <span class="review"></span></p><p>-2log0.5 = log <input type="text" style="height: auto;" data-c="4"> <span class="review"></span></p></div><div class="q-explanation"><p>This question uses the three log laws:</p><p>loga + logb = logab</p><p>loga - log b = log<span class="math-tex">\(\frac{a}{b}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p><span class="math-tex">\(\log _{ 3 }{ 5 } =\frac { \log _{ c }{ a } }{ \log _{ c }{ b } } \)</span></p><p>What are the values of <strong><em>a </em></strong>and <strong><em>b</em></strong>?</p></div><div class="q-answer"><p><em><strong>a</strong></em> = <input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p><em><strong>b</strong></em> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>This question uses the change of base formula:</p><p><span class="math-tex">\(\log _{ b }{ a } =\frac { \log _{ c }{ a } }{ \log _{ c }{ b } } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Which is the correct answer to log<sub>5</sub>10?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> <span class="math-tex">\(1\over log5\)</span></label><label class="radio"> </label></p><p><label class="radio"><input type="radio"> log2</label></p><p><label class="radio"><input type="radio"> 2</label></p><p><label class="radio"><input type="radio"> <span class="math-tex">\(\frac{log5}{log10}\)</span></label></p></div><div class="q-explanation"><p><span class="math-tex">\(\log _{ 5}{ 10 } =\frac { \log _{ 10 }{ 10 } }{ \log _{ 10 }{ 5 } } =\frac {1 }{ \log { 5 } } \)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The first term of a geometric sequence is ln2 and the common ratio is 5.</p><p>What is the second term?</p></div><div class="q-answer"><p>Second term = ln <input type="text" style="height: auto;" data-c="32"> <span class="review"></span></p></div><div class="q-explanation"><p>Second term = 5ln2 = ln2<sup>5</sup></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>Here is an arithmetic sequence</p><p>ln2 , U<sub>2</sub> , ln50</p><p>What is the common difference, <em><strong>d</strong></em> and the second term <em><strong>U<sub>2</sub></strong></em>?</p></div><div class="q-answer"><p><strong><em>d</em></strong> = ln<input type="text" style="height: auto;" data-c="5"> <span class="review"></span></p><p>U<sub>2</sub> = ln <input type="text" style="height: auto;" data-c="10"> <span class="review"></span></p></div><div class="q-explanation"><p>2d = ln50-ln2</p><p>2d = ln25</p><p>d = 0.5ln25</p><p>d = ln25<sup>0.5</sup></p><p>d = ln5</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graphs of y = f(x), y = x and y = f <sup>-1</sup>(x) are plotted below, where f(x) = 3<sup>x</sup></p><p><img alt="" src="../../files/algebra/logarithms/esq.jpg" style="width: 400px; height: 318px;"></p><p>What is the function f <sup>-1</sup>(x)?</p></div><div class="q-answer"><p><label class="radio"><input class="c" type="radio"> log<sub>3</sub>x</label></p><p><label class="radio"><input type="radio"> (log<sub>3</sub>x)<sup>-1</sup> </label></p><p><label class="radio"><input type="radio"> 3logx</label></p><p><label class="radio"><input type="radio"> log3<sup>x</sup></label></p></div><div class="q-explanation"><p>The inverse of a<sup>x</sup> is log<sub>a</sub>x</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The graph of y = log<sub>2</sub>x is plotted below. What is the value of <strong><em>a</em></strong>?</p><p><img alt="" src="../../files/algebra/logarithms/esq9.jpg" style="width: 400px; height: 272px;"></p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p></div><div class="q-explanation"><p>log<sub>2</sub>8=3</p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div><div class="exercise shadow-bottom"><div class="q-question"><p>The solution to the equation 3<sup>x</sup> = 2<sup>x+1</sup> can be written in the form <span class="math-tex">\(\frac{log\ a}{log\frac{b}{c}}\)</span></p><p>Find <strong><em>a</em></strong> , <strong><em>b</em></strong> and <strong><em>c</em></strong>.</p></div><div class="q-answer"><p><strong><em>a</em></strong> = <input type="text" style="height: auto;" data-c="2"> <span class="review"></span></p><p><strong><em>b</em></strong> = <input type="text" style="height: auto;" data-c="3"> <span class="review"></span></p><p><strong><em>c </em></strong>= <input type="text" style="height: auto;" data-c="2"> <span class="review"></span> </p></div><div class="q-explanation"><p>3<sup>x</sup> = 2<sup>x+1</sup></p><p>log3<sup>x</sup> = log2<sup>x+1</sup></p><p>xlog3 = (x+1)log2</p><p>xlog3 = xlog2 = log2</p><p>xlog3 - xlog2 = log2</p><p>x(log3 - log2) = log2</p><p>x = <span class="math-tex">\(\frac{log2}{log3-log2}\)</span></p><p>x = <span class="math-tex">\(\frac{log2}{log\frac{3}{2}}\)</span></p></div><div class="slide-q-actions"><button class="btn btn-default btn-sm btn-xs-block text-xs-center check"><i class="fa fa-check-square-o"></i> Check</button></div></div> </div> </div> <div class="modal-footer slide-quiz-actions"> <div class=""> <div class="pull-left pull-xs-none mb-xs-3"> <button class="btn btn-default d-xs-none btn-prev"> <i class="fa fa-arrow-left"></i> Prev </button> </div> <div class="pull-right pull-xs-none"> <button class="btn btn-success btn-xs-block text-xs-center btn-results" style="display: none"> <i class="fa fa-bar-chart"></i> Check Results </button> <button class="btn btn-default d-xs-none btn-next"> Next <i class="fa fa-arrow-right"></i> </button> <button class="btn btn-default btn-xs-block text-xs-center btn-close" data-dismiss="modal" style="display: none"> Close </button> </div> </div> </div> </div> </div></div> </div> <div class="panel-footer"> <div> <p>text</p> </div> </div> </div> <div class="panel panel-has-colored-body panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Exam-style Questions</p> </div> </div> <div class="panel-body"> <div class="panel panel-has-colored-body panel-default panel-has-border"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 1</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="463"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL easy"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Find the value of each of the following, giving your answer as an integer</p> <p>a. <span class="math-tex">\({ log }_{ 4 }16\)</span></p> <p>b. <span class="math-tex">\({ log }_{ 4 }2+{ log }_{ 4 }32\)</span></p> <p>c. <span class="math-tex">\({ log }_{ 4 }8-{ log }_{ 4 }32\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>The log laws are useful</p> <p><content><span class="math-tex">\({ log }a+logb=logab\\ { log }a-logb=log\frac { a }{ b } \)</span></content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/esq1_logs_and_indices.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/esq1_logs_and_indices.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 2</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="464"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>a. Given that 3<sup>a</sup> =27 , write down the value of <strong><em>a</em></strong></p> <p>b. Hence of otherwise solve 27<sup>x+4</sup> = 9<sup>3x+1</sup></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden">b. Replace 27 by 3<sup>a</sup> and 9 by 3²<content></content></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/esq2_logs_and_indices.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/esq2_logs_and_indices.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-has-border panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 3</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="465"> <p><img class="sibico" src="../../../img/sibico/hl-green.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL easy"> <img class="sibico" src="../../../img/sibico/sl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL moderate"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Solve the equation log<sub>2</sub>(x - 3) = 1 - log<sub>2</sub>(x - 4)</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Rearrange the equation to have the logs on one side.</p> <p><content>Change the log equation into an index equation.</content></p> <p>You will need to solve a quadratic equation.</p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/esq3_logs_and_indices.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/esq3_logs_and_indices.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-has-border panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 4</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="466"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Solve the equation <span class="math-tex">\(3^{ x-1 }=\frac { 2 }{ { 4 }^{ x+1 } } \)</span> , giving your answer in the form <span class="math-tex">\(x=\frac{lna}{lnb}\)</span> , where <strong><em>a </em></strong>and <strong><em>b</em></strong> are rational numbers.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>Take logs (base e) of both sides.</p> <p><content>Rearrange the equation to gather xs on one side.</content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/esq4_logs_and_indices.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/esq4_logs_and_indices.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 5</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="467"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>Solve <span class="math-tex">\(ln(sinx)−ln(cosx)=e\)</span> for <span class="math-tex">\(0<x<2\pi\)</span></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p>In this question, you have to solve a trigonometric equation.</p> <p>This identity is useful <span class="math-tex">\(tanx = \frac{sinx}{cosx}\)</span><content> </content></p> </section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/esq5_logs_and_indices.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/esq5_logs_and_indices.pdf" width="640"></iframe></p> </section> </div> <p> </p> </div> </div> <div class="panel-footer"> <div> <p> </p> </div> </div> </div> <div class="panel panel-has-colored-body panel-expandable panel-has-border panel-default"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 6</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="468"> <p><img class="sibico" src="../../../img/sibico/hl-orange.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL moderate"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p><span class="math-tex">\(a=\log _{ 2 }{ 2+\log _{ 2 }{ \frac { 3 }{ 2 } + } } \log _{ 2 }{ \frac { 4 }{ 3 } + } \quad ...\quad +\log _{ 2 }{ \frac { 32 }{ 31 } } \)</span></p> <p>Given that <span class="math-tex">\(a\in \mathbb{Z}\)</span> , find the value of <strong><em>a</em></strong></p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"><content> </content><span class="math-tex">\(\log _{ 2 }{ \frac { 3 }{ 2 }}=\log _{ 2 }3-\log _{ 2 }2\)</span></section> <h4><span class="fa fa-pencil" style="color:rgb(0, 0, 0);"></span> Full Solution</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><span class="fa fa-print" style="color:rgb(0, 0, 0);font-size:14px;"></span> Print from <a href="../../files/algebra/logarithms/esq6_logs_and_indices.pdf" target="_blank">here</a></p> <p><iframe align="middle" frameborder="0" height="480" scrolling="yes" src="../../files/algebra/logarithms/esq6_logs_and_indices.pdf" width="640"></iframe></p> </section> </div> </div> </div> <div class="panel-footer"> <div> </div> </div> </div> <div class="panel panel-has-colored-body panel-default panel-has-border panel-expandable"> <div class="panel-heading"><a class="expander" href="#"><span class="fa fa-plus"></span></a> <div> <p>Question 7</p> </div> </div> <div class="panel-body"> <div> <div class="smart-object center" data-id="469"> <p><img class="sibico" src="../../../img/sibico/hl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="HL difficult"> <img class="sibico" src="../../../img/sibico/sl-red.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="SL difficult"> <img class="sibico" src="../../../img/sibico/no-calc.svg" style="height:1.25em;width: auto;vertical-align:text-bottom" title="No calculator"></p> <p>The first three terms of a geometric sequence are <span class="math-tex">\(\log _{ 3 }{x\quad ,\quad \log _{ 9 }{ x\quad ,\quad \log _{ 81 }{ x } } } \)</span></p> <p>Find the value of <strong>x</strong> if the sum to infinity is 8.</p> <h4><span class="fa fa-support" style="color:rgb(0, 0, 0);"></span> Hint</h4> <button class="btn btn-xs bg-turquoise showhider"><i class="fa fa-fw fa-plus"></i></button><section class="hiddenbox hidden"> <p><content> </content>The common ratio is <span class="math-tex">\(\frac { \log _{ x }{ 9 } }{ \log _{ x }{ 3 } } \)</span></p> <p>The change of base formula is useful <span class="math-tex">\(\log _{ b }{ a } =\frac { \log _{ c }{ a } }{ \log _{ c }{ b } } \)</span></p> </section> <h4><span class="fa fa-pencil" 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