Question

can someone explain this:
if f(x) = log 1+x/1-x (-1<x<1)
show that f(a)+ f(b) = f(a+b/1+ab)
(|a|<1, |b|<1)

Answer 1

If f(x)=log1−x1+x,then

f(a)+f(b)=log1−a1+a+log1−b1+b.

We can use that logx+logy=logxy and logx−logy=logxy. Then

f(a)+f(b)=log(1−a)−log(1+a)+log(1−b)−log(1+b)

f(a)+f(b)=log(1−a)(1−b)−log(1+a)(1+b).

Alternatively.

f(a)+f(b)=log(1−a)(1−b)(1+a)(1+b).