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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
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
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:


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