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  4. If f(x)= log ( (1+x/1-x) ),-1<x<1, then f( (3x+x3/1+3x2) )-f( (2x/1+x2) ) is

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Q. If $ f(x)=\log \left( \frac{1+x}{1-x} \right),-1
KEAMKEAM 2008

Solution:

Given, $ f(x)=\log \left( \frac{1+x}{1-x} \right) $
$ \therefore $ $ f\left( \frac{3x+{{x}^{3}}}{1+3{{x}^{2}}} \right)-f\left( \frac{2x}{1+{{x}^{2}}} \right) $
$=\log \left( \frac{1+\left( \frac{3x+{{x}^{3}}}{1+3x2} \right)}{1-\left( \frac{3x+{{x}^{3}}}{1+3{{x}^{2}}} \right)} \right)-\log \left( \frac{1+\frac{2x}{1+{{x}^{2}}}}{1-\frac{2x}{1+{{x}^{2}}}} \right) $
$=\log {{\left( \frac{1+x}{1-x} \right)}^{3}}-\log {{\left( \frac{1+x}{1-x} \right)}^{2}} $
$=\log \left( \frac{1+x}{1-x} \right) $
$=f(x) $