Question

The inverse of $y=5^{\log x}$ is :

• A. $x=5^{\text{logy }}$
• B. $x=y^{\text{log5}}$
• C. $x=y^{\frac{1}{\log 5}}$
• D. $x=5^{\frac{1}{\log y}}$

Answer 1

Answer: (C)

Given $y=5^{\left({\log }_{a}x\right)}=f(x)$
Interchanging $x\&y$ for inverse
$x=5^{\left({\log }_{a}y\right)}=y^{\left({\log }_{a}5\right)}$
option (1) or option (2)
Further, from given relation
${\log }_{5}y={\log }_{a}x$
$\Rightarrow x=a^{\left({\log }_{5}y\right)}=y^{\left({\log }_{5}a\right)}$
$\Rightarrow x=y^{\left(\frac{1}{{\log }_{a}5}\right)}=f^{-1}(y)$