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  4. If f(x)= log e((1+x/1-x)), then f((2 x/1+x2)) is equal to

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Q. If $f(x)=\log _{e}\left(\frac{1+x}{1-x}\right)$, then $f\left(\frac{2 x}{1+x^2}\right)$ is equal to

Relations and Functions

Solution:

We have, $f(x)=\log _e\left(\frac{1+x}{1-x}\right)$
$f\left(\frac{2 x}{1+x^2}\right) =\log _e\left(\frac{1+\frac{2 x}{1+x^2}}{1-\frac{2 x}{1+x^2}}\right) $
$ =\log _e\left(\frac{1+x^2+2 x}{1+x^2-2 x}\right) $
$ =\log _e\left(\frac{1+x}{1-x}\right)^2$
$=2 \log _e\left(\frac{1+x}{1-x}\right)$
$\left(\because \log a^b=b \log a\right)$
$ =2 f(x)$