Proving Logarithmic identity
By Sarah Scott •
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 ?
$\endgroup$ 101 Answer
$\begingroup$Just so this doesn't remain unanswered:
This is a statement of the fact that the functions $a^x$ and $\log_a\;x$ are inverses of each other; thus, $\log_a\;a^x=a^{\log_a\;x}=x$
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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