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Today, I read these two new Logarithmic identity $\displaystyle$ $$a^{\log_a m} = m $$ $$\log_{a^q}{m^p} = \frac{p}{q} \log_a m$$ Both of them seems new to me,so even after solving some problems (directly) based on thesm I haven't fully understood how they holds good,Could anybody show me how to prove them ?

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You know that $y = \log_b x \iff b^y = x$, right? –  Djaian Nov 18 '10 at 9:57
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To restate Djaian's hint: the logarithm and exponential are inverses of each other. –  J. M. Nov 18 '10 at 10:03
    
Ok.That is for the first one I got it now ;) how about the second one ? –  Quixotic Nov 18 '10 at 10:06
    
For the second identity: you're aware of the "change-of-base" formula, yes? –  J. M. Nov 18 '10 at 10:07
    
Edit to $a^{\log_am}=m$ –  Américo Tavares Nov 18 '10 at 10:08

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up vote 5 down vote accepted

Just so this doesn't remain unanswered:

  1. This is a statement of the fact that the functions $a^x$ and $\log_a\;x$ are inverses of each other; thus, $\log_a\;a^x=a^{\log_a\;x}=x$

  2. Letting c be a positive real number not equal to one:

$\log_{a^q}\;m^p=\frac{\log_c\;m^p}{\log_c\;a^q}=\frac{p\;\log_c\;m}{q\;\log_c\;a}=\frac{p}{q}\log_a\;m$

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