Question

Find the inverse of the following functions:

$f\left(x\right)=5^{{\log }_{e}x},x>0$, then $f^{-1}\left(1\right)=$

Answer 1

$f\left(x\right)=5^{\log x}=y$ (say)
Take log both sides on base $5$
$\Rightarrow {\log }_{e}x={\log }_{5}y\Rightarrow x=e^{{\log }_{5}y}=f^{-1}\left(y\right)$
$\Rightarrow f^{-1}\left(1\right)=e^{{\log }_{5}1}=e^0=1$