I am wondering about a proof of additivity of natural logarithm $$\ln(ab)=\ln a+\ln b$$ using the power series $$\ln(x)=\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}}{n}(x-1)^n.$$ Since this power series converges only in $x\in(0, 2],$ usually this does not use as "the" definition.
Is there any (nice) way to prove this using only the power series (not exponential function)?
One possible idea is invert the power series to get another power series $p(x)$ that converges everywhere and has the property $p(x+y)=p(x)p(y)$ (via Cauchy products) and use it to derive the desire result. But this is not what I am looking for. Because I do not want exponentials anywhere in the proof. Probably something analogous to Cauchy products would do the job.