Question

Consider the function $y={\log }_a(x+\sqrt{x^2+1}),a>0,a\ne 1$. The inverse of the function

• A. does not exist
• B. is $x={\log }_a(y+\sqrt{y^2+1})$
• C. is $x=\sin \left(y\ln a\right)$
• D. is $x=\cos h(-y\ln \frac{1}{a})$

Answer 1

Answer: (C)

is $x=\sin \left(y\ln a\right)$

$a^y=(x+\sqrt{x^2+1})\Rightarrow a^{-y}=\sqrt{x^2+1}-x\Rightarrow a^y-a^{-y}=2x$

$\Rightarrow f^{-1}\left(y\right)=x=\frac{a^y-a^{-y}}{2}=\frac{e^{y\ln a}-e^{-y\ln a}}{2}=\sin \left(y\ln na\right)(\sin h\left(x\right)=\frac{e^x-e^{-x}}{2})$