Logarithmic property justification

Question

I saw this particular line slammed in a proof and it bothers me I can't understand why this is obvious and how would one justify this : $$ 7^{\log (n)} = n^{\log (7)} $$

Can anyone explain ?

Answer 1

$$7^{\ln n}=\exp (\ln 7 \ln n)=n^{\ln 7}.$$

Answer 2

$$7^{\log n}=7^{\frac{\log_7(n)}{\log_7(10)}}=7^{\log_7(n^{1/\log_7(10)})}=n^{1/\log_7(10)}=n^{\log_7(7)/\log_7(10)}=n^{\log(7)}$$

Answer 3

I think everyone is missing the point that the definition for $x^\alpha$ for irrational $\alpha$ is

$$x^\alpha =e^{\alpha\ln x}.$$

This is why the other answers make sense.

Answer 4

Take $\log$ of the LHS

\begin{equation} \begin{aligned} \log{(7^{\log{n}}}) =& (\log{n})(\log{7}) \\ =& \log{(n^{\log{7}}}) \end{aligned} \end{equation}

Then removing the log we took earlier we can clearly see $7^{\log{n}} = n^{\log{7}}.$

Hope this helps.