TITLE: Infinite sum of logs puzzle
QUESTION [8 upvotes]: Here is a neat infinite sum puzzle:

Prove the following is true when $|x|<1$
$$-\ln(1-x)=\ln(1+x)+\ln(1+x^2)+\ln(1+x^4)\dots\ln(1+x^{2^k})\dots\\-\ln(1-x)=\sum_{k=0}^\infty\ln(1+x^{2^k})$$

Hope you all enjoy!

 HINT: The answer is probably simpler than you might think.

REPLY [8 votes]: Exponentiate both sides to get $\frac{1}{1-x} = (1+ x)(1 + x^2)$... By uniqueness of the binary representation of a nonnegative integer we find that the $x^n$ coefficient on the right side is $1$ for all $n$, so equality follows by power series representation of the left side.