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Question Number 196596 by MATHEMATICSAM last updated on 27/Aug/23

If f(x) = ln(((1 + x)/(1 − x))) then prove that f(((2x)/(1 + x^2 ))) = 2f(x).

$$\mathrm{If}\:{f}\left({x}\right)\:=\:\mathrm{ln}\left(\frac{\mathrm{1}\:+\:{x}}{\mathrm{1}\:−\:{x}}\right)\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$${f}\left(\frac{\mathrm{2}{x}}{\mathrm{1}\:+\:{x}^{\mathrm{2}} }\right)\:=\:\mathrm{2}{f}\left({x}\right). \\ $$

Commented by mokys last updated on 27/Aug/23

f (((2x)/(1+x^2 )))= ln(((1+((2x)/(1+x^2 )))/(1−((2x)/(1+x^2 )))) ) = ln ( ((1+2x+x^2 )/(1−2x+x^2 ))) = ln ( ((1+x)/(1−x)))^2 = 2 ln (((1+x)/(1−x))) = 2 f(x)

$${f}\:\left(\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }\right)=\:{ln}\left(\frac{\mathrm{1}+\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }}{\mathrm{1}−\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }}\:\right)\:=\:{ln}\:\left(\:\frac{\mathrm{1}+\mathrm{2}{x}+{x}^{\mathrm{2}} }{\mathrm{1}−\mathrm{2}{x}+{x}^{\mathrm{2}} }\right)\:\:\: \\ $$$$ \\ $$$$\:=\:{ln}\:\left(\:\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right)^{\mathrm{2}} \:=\:\mathrm{2}\:{ln}\:\left(\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}\right)\:=\:\mathrm{2}\:{f}\left({x}\right) \\ $$

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