Question

If $f:R^{+}\to R$ such that $f\left(x\right)={\log }_{5}x$ then $f^{-1}\left(x\right)=$

• A. ${\log }_{x}10$
• B. $5^x$
• C. $3^{-x}$
• D. $3^{1/x}$

Answer 1

The correct option is B $5^x$

$f\left(x\right)=y$

$\Rightarrow {\log }_{5}x=y$

$\Rightarrow \frac{\log x}{\log 5}=y$

$\Rightarrow \log x=y\log 5$

$\Rightarrow e^{\log x}=e^{y\log 5}$

$\Rightarrow e^{\log x}=e^{\log 5^y}$

$[\because a\log x=\log x^a$]

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

$\therefore f^{-1}\left(x\right)=5^x$