Question

The inverse of $y=5^{\log \left(x\right)}$ is

• A. $x=5^{\log \left(y\right)}$
• B. $x=5^{\log \left(y\right)}c$
• C. $x=y^{\frac{1}{\log \left(5\right)}}$
• D. $x=5^{\frac{1}{\log \left(y\right)}}$

Answer 1

The correct option is C

$x=y^{\frac{1}{\log \left(5\right)}}$

Explanation for the correct option:

Simplify the equation in terms of $y$

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

Taking log both sides

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

Let the base of log be $5$

${\log }_5\left(y\right)={\log }_e\left(x\right)$ $\left[\because {\log }_a\left(a\right)=1\right]$

Taking anti-log

$$\begin{gathered} e^{{\log }_5\left(y\right)}=x \\ \Rightarrow y^{{}^{{\log }_5\left(e\right)}}=x \\ \Rightarrow y^{\frac{1}{\log 5}}=x \end{gathered}$$ $\left[\because b^{{\log }_e\left(a\right)}=a^{{\log }_e\left(b\right)}\right]$

Hence, option C is correct.