Question

The inverse of the function $f\left(x\right)=2^{x/(x-1)}$ is

• A. $\frac{{\log }_{2}x}{{\log }_{2}x-1}$
• B. $\frac{{\log }_{2}x-1}{{\log }_{2}x}$
• C. $\frac{{\log }_{2}x}{{\log }_{2}x+1}$
• D. $\frac{{\log }_{2}x+1}{{\log }_{2}x-1}$

Answer 1

Answer: (A)

$\frac{{\log }_{2}x}{{\log }_{2}x-1}$

$f\left(x\right)=2^{x/(x-1)}$
Put $y=f\left(x\right)$
$y=2^{x/(x-1)}$
${\log }_{2}y=\frac{x}{x-1}$
$x{\log }_{2}y-{\log }_{2}y=x$
$x({\log }_{2}y-1)={\log }_{2}y$
$x=\frac{{\log }_{2}y}{{\log }_{2}y-1}$
$f^{-1}\left(y\right)=\frac{{\log }_{2}y}{{\log }_{2}y-1}$
$f^{-1}\left(x\right)=\frac{{\log }_{2}x}{{\log }_{2}x-1}$