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      <text>4.2.16 (Changing base)</text>
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      <text><![CDATA[<h1>Change of base formula</h1>

<p>It is sometimes required to express the logarithm with respect to one base in terms of a logarithm with respect to another base.</p>
<p>Now \(b=a^x\) implies \(\log_a b=x\) where we have used logs to base \(a\). If for some reason we want to use another base, \(p\) say, then we do as follows.</p> 
<p>We take logs to base \(p\) of both sides of \(b=a^n\), which gives
\(\log_p(b)= \log_p(a^n)=n\log_pa\) (using one of the logarithm laws), so \(n=\dfrac{\log_p(b)}{\log_p(a)}\). But we also have 
\(\log_ab=n\).</p>
<p>Therefore, we conclude \(\log_ab=\dfrac{\log_p(b)}{\log_p(a)}.\) This is the rule to be used when converting logarithms from one base to another.
\[
\log_ab=\dfrac{\log_p(b)}{\log_p(a)}.
\]
In particular for base \(10\) and natural logs: 
\[
\log_ab=\dfrac{\log (b)}{\log (a)} \qquad \mbox { and } \qquad \log_ab=\dfrac{\ln (b)}{\ln (a)}.
\]
</p>
<br/>
<h3>Example</h3>

<p>Use the change of base formula to change the base of \(\log_3(7)\) to base \(10\) and base \(e\), thus evaluate this log.

<h3>Solution</h3>

<p>Using the formula changing to base \(10\) we have,
\[
\log_37=\dfrac{\log 7}{\log 3}=\dfrac{0.8450980}{0.4771212}=1.7712437
\] 
Alternatively, 
\[
\log_37=\dfrac{\ln 7}{\ln 3}=\dfrac{1.9459101}{1.0986123}=1.7712437
\]
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