Proof of natural log identities I need to prove a few of the following identities from a real analysis perspective- this means I do not have access the $\ln e^2 = 2$ type definition of the log function. I am developing the log function from the definition $log x = \int_1^x \frac1t \mathrm dt$ for $0 < x$.
I need to prove that: $\ln(ab) = \ln(a) + \ln(b)$ first. I imagine the proof for that will apply itself pretty directly to $\ln(x^n) = n\ln(x)$. I've been given the hint that the idea here is to define $f(x) = \ln(ax)$ and show that $f'(x) = \ln(x)'$, implying $f(x) = L(ax) = L(x) + k$, where I can show that $k = L(a)$. 
Any help with starting this?

Shorter version
$$\ln(ab)=\int_1^{ab}\frac{1}{t}dt=\int_b^{ab}\frac{1}{t}dt+\int_1^{b}\frac{1}{t}dt=\int_1^{a}\frac{1}{u}du+\int_1^{b}\frac{1}{t}dt=\ln(a)+\ln(b)$$
where I made the substitution $u=bt$.
