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Math Help - Logarithm Proof

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    Logarithm Proof

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    I have no idea where to begin. I know some of the property rules of logarithms but, that did not help. Any ideas?
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    Hello,
    Quote Originally Posted by chrozer View Post
    I have no idea where to begin. I know some of the property rules of logarithms but, that did not help. Any ideas?
    Note that \log_a x=\frac{\ln(x)}{\ln(a)}

    So \log_{\frac ab} x=\frac{\ln(x)}{\ln \left(\tfrac ab\right)}

    but \ln \frac ab=\ln a-\ln b (basic log property)

    Hence :


    \frac{\log_a x}{\log_{\frac ab} x}=\frac{\ln(x)}{\ln(a)} \cdot \frac{\ln(a)-\ln(b)}{\ln(x)}=1-\frac{\ln(b)}{\ln(a)}=1-\log_a b=1+\log_a \left(\frac 1b\right)

    (using the rule -log(b)=log(1/b))
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    Thanks alot.
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