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      <text>4.2.17 (Changing base example)</text>
    </name>
    <questiontext format="html">
      <text><![CDATA[<h3>Example</h3>

<p>Solve for \(x\) if \(3\ln x+4\log x=1\).</p>

<h3>Solution</h3>

<p>This has logs to two different bases. So we must first express each logarithm in terms of logs to the <i>same</i> base, we will use base \(e\).Recall</p>
\[
\log x=\dfrac{\ln x}{\ln 10}.
\] 
So \(3\ln x+4\log x=1\) becomes
\[ 
3\ln x+4\dfrac{\ln x}{\ln 10}=1\qquad\text{or}\qquad \left(3+\dfrac{4}{\ln 10}\right)\ln x=1
\]
leading to \(\ln x=\dfrac{\ln 10}{3\ln 10 +4}=\dfrac{2.302585}{10.907755}=0.211096\) and so
\[ 
x=e^{0.211096}=1.2350311.
\]]]></text>
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