The family of recurrence considered is of the form
$$ T(n) = aT(n/b) + n^c $$ $a,b,c$ are integers.
One case of master theorem states:
if $c< log_b a$, then $T(n) = \Theta(n^{log_b a}) $. I have a doubt in a specific line of the proof for this case, which requires a specific log-property.
$$\begin{align} n^c \left(\frac{a}{b^c}\right)^{log_b n} & = n^c \frac{a^{log_b n}}{(b^c)^{log_b n}} \\ & = n^c \frac{n^{log_b a}}{n^{log_b b^c}} \\ & = n^c \frac{n^log_b a}{n^c} \\ & = n^{log_b a} \end{align} $$
My problem is in step 2, where $a^{log_b n}$ becomes $n^{log_b a}$.
When I tried out the proof, the final answer I ended up with was $a^{log_b n}$
What property of logarithms, am I missing out here. Thanks in advance.
Using the rule for logarithm of a power, $$\log_b(a^{\log_bn})=(\log_bn)(\log_ba)=(\log_ba)(\log_bn)=\log_b(n^{\log_ba})\ .$$ So $$a^{\log_bn}=n^{\log_ba}\ .$$