The inverse of y=5^ x is - Vidyadip

MCQ

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

A
$x =5^{\text {logy }}$
B
$x=y^{\log 5}$
✓
$x = y ^{\frac{1}{\log 5}}$
D
$x =5^{\frac{1}{\log y}}$

✓

Answer

Correct option: C.

$x = y ^{\frac{1}{\log 5}}$

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

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

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

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

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