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
$\log _{5} y=\log _{a} x$
$\Rightarrow x = a ^{\left(\log _{5} y \right)}=y ^{\left(\log _{5} a \right)}$
$\Rightarrow x = y ^{\left(\frac{1}{\log _{a} 5}\right)}=f^{-1}( y )$