The inverse of

Question:

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

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

  2. $x=y^{\log 5}$

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

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


Correct Option: , 3

Solution:

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

Interchanging $x$ \& $y$ for inverse

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

option (1) or option (2)

Further, from given relation

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

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

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

option (3)

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