Inverse of y=5^{log x} is

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1.Relation and Function

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

A

$x =5^{\text {logy }}$

B

$x=y^{\log 5}$

C

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

D

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

(JEE MAIN-2021)

Solution

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

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

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

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

Standard 12

Mathematics

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