Question

Find whether $\log (x+\sqrt{1+x^2})$ is even or odd

Answer 1

Let $f\left(x\right)=\log (x+\sqrt{1+x^2})$
$\Rightarrow f(-x)=\log (\sqrt{1+x^2}-x)$
Rationalizing,
$f(-x)=\log \left[\right(\sqrt{1+x^2}-x).\frac{\sqrt{1+x^2}+x}{\sqrt{1+x^2}+x}]$

$=\log \left[\frac{1}{\sqrt{1+x^2}+x}\right]$

$\Rightarrow f(-x)=-f\left(x\right)$
Thus $f\left(x\right)$ is an odd function.