<p>Logarithms turn powers and exponentials into linear relationships. All logarithm arguments must be positive.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 580px;"><colgroup><col style="width: 15.5172%;"><col style="width: 31.0345%;"><col style="width: 22.4138%;"><col style="width: 13.7931%;"><col style="width: 17.2414%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Law</strong></th><th style="text-align: center; --cell-justify: center"><strong>Take logarithms</strong></th><th style="text-align: center; --cell-justify: center"><strong>Plot</strong></th><th style="text-align: center; --cell-justify: center"><strong>Gradient</strong></th><th style="text-align: center; --cell-justify: center"><strong>Vertical intercept</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(y=ax^n\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(y=ab^x\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(y=Ae^{kx}\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<div class="content-block content-block--columns"><div class="question-columns question-columns--2" data-layout="2"><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="210" data-height="140" data-proportional style="width:210px;height:140px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:19px;--graph-width:210px;--graph-aspect:210 / 140;"></div><p>\(y=ax^n\): the graph of \(\lg y\) against \(\lg x\) is a straight line with gradient \(n\) and intercept \(\lg a\).</p></div><div class="question-column" style="flex: 1; align-self: flex-start;"><div class="graph-block" data-graph-config="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" data-width="210" data-height="140" data-proportional style="width:210px;height:140px;max-width:100%;display:flow-root;margin:0 auto;margin-bottom:13px;--graph-width:210px;--graph-aspect:210 / 140;"></div><p>\(y=ab^x\): the graph of \(\lg y\) against \(x\) is a straight line with gradient \(\lg b\) and intercept \(\lg a\).</p></div></div></div>
<p class="paragraph-header">Recovering constants</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\lg a=c\iff a=10^c\qquad\ln A=c\iff A=e^c\)</p></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 480px;"><colgroup><col style="width: 50.0000%;"><col style="width: 50.0000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Read from the graph</strong></th><th style="text-align: center; --cell-justify: center"><strong>Undo the logarithm</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Vertical intercept \(=\lg a\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Gradient \(=\lg b\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">Vertical intercept \(=\ln A\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>
<p>Use the same logarithm base as the graph.</p>

Logarithms turn powers and exponentials into linear relationships. All logarithm arguments must be positive.

\(\begin{aligned} \lg y&=\lg a+n\lg x \\ \lg y&=n\lg x+\lg a \end{aligned}\)

\(\begin{aligned} \lg y&=\lg a+x\lg b \\ \lg y&=(\lg b)x+\lg a \end{aligned}\)

\(\begin{aligned} \ln y&=\ln A+kx \\ \ln y&=kx+\ln A \end{aligned}\)

\(\lg a=c\iff a=10^c\qquad\ln A=c\iff A=e^c\)

Use the same logarithm base as the graph.

Law	Take logarithms	Plot	Gradient	Vertical intercept
\(y=ax^n\)
\(y=ab^x\)
\(y=Ae^{kx}\)

Straightening Curves with Logarithms

Straightening Curves with Logarithms

Read from the graph	Undo the logarithm
Vertical intercept \(=\lg a\)
Gradient \(=\lg b\)
Vertical intercept \(=\ln A\)