# Math Help - Logarithm Proof

1. ## 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?

2. Hello,
Originally Posted by chrozer
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))

3. Thanks alot.