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      <text>6.3.4.1 Changing base in logarithms</text>
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      <text><![CDATA[<h3>Changing base in logarithms</h3><p></p>
<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 \(b=a^n\) implies \(\log_ab=n\) 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 \(\log_ab=\dfrac{\log_p(b)}{\log_p(a)}.\)</p>
<p>This is the rule to be used when converting logarithms from one base to another.</p>
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<h4>Key Point</h4>
<p>\[\log_ab=\dfrac{\log_p(b)}{\log_p(a)}.\]</p>
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<p></p>
<p>For base 10 logs: \[\log_ab=\dfrac{\log (b)}{\log (a)}.\]</p>
<p>For natural logs: \[\log_ab=\dfrac{\ln (b)}{\ln (a)}.\]</p>
<hr>
<h4 class="HELM_example">Example&nbsp;</h4><p>For example, \(\log_37\) cannot be determined directly on your calculator since logs to base 3 are not available but it can be found using the above formulas (changing to base \(10\) or base 
\(e\)).
\[\log_37=\dfrac{\log 7}{\log 3}=\dfrac{0.8450980}{0.4771212}=1.7712437\] 
(Check, on your calculator, that \(3^{1.7712437}=7\)).</p>
<p>Alternatively, \[\log_37=\dfrac{\ln 7}{\ln 3}=\dfrac{1.9459101}{1.0986123}=1.7712437\]
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