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I am looking for a proof of the fact $\ln(ab)=\ln a+\ln b$ 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 a definition.
Is there any (nice) way to prove this?

Thank you.

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    It's easy to prove using power series that $\exp(a+b) = \exp(a) \exp (b)$, which would amount to the same thing. – 2017-01-15
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    I am looking for a direct proof without using Exponentiation. – 2017-01-15
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    The given series converges for $x\in(0,2]$. – 2017-01-15
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    @egreg: Ohh.. Thank you. – 2017-01-15

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