JEE MAIN - Mathematics (2021 - 17th March Morning Shift - No. 7)
The inverse of $$y = {5^{\log x}}$$ is :
$$x = {5^{\log y}}$$
$$x = {y^{{1 \over {\log 5}}}}$$
$$x = {5^{{1 \over {\log y}}}}$$
$$x = {y^{\log 5}}$$
Explanation
$$y = {5^{\log x}}$$
$$ \Rightarrow \log y = \log x.log5$$
$$ \Rightarrow \log x = {{\log y} \over {\log 5}} = {\log _5}y$$
$$ \Rightarrow x = {e^{{{\log }_5}y}}$$
$$ \Rightarrow x = {y^{{{\log }_5}e}}$$
$$ \Rightarrow x = {y^{{1 \over {\log 5}}}}$$
$$ \Rightarrow \log y = \log x.log5$$
$$ \Rightarrow \log x = {{\log y} \over {\log 5}} = {\log _5}y$$
$$ \Rightarrow x = {e^{{{\log }_5}y}}$$
$$ \Rightarrow x = {y^{{{\log }_5}e}}$$
$$ \Rightarrow x = {y^{{1 \over {\log 5}}}}$$
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