Prove $\log_{a} c + \log_{b} c = \log_{a+b} c$ if and only if $1 + \log_{b} a = \log_{a+b} a$ If a, b and c are positive numbers, than equality $$\log_{a} c + \log_{b} c = \log_{a+b} c$$ is true if and only if $$1 + \log_{b} a = \log_{a+b} a$$
Prove it!
I have looked at the solution but it is not clear for me.
We will prove that if $$\log_{a} c + \log_{b} c = \log_{a+b} c$$
than it is 
$$1 + \log_{b} a = \log_{a+b} a$$ 
$$\log_{a} c + \log_{b} c = \frac{\log_{a+b} c}{\log_{a+b} a} +\frac{\log_{a+b} c}{\log_{a+b} b} =\log_{a+b} c $$
So this is only thing that is not clear for me how is this equal.
$$\frac{\log_{a+b} c}{\log_{a+b} a} +\frac{\log_{a+b} c}{\log_{a+b} b} =\log_{a+b} c $$

WE have:
$$1+  log_b (a)= log_{a+b} (a)  $$
Let $a=c$ then:
$$ log _a(c)= log _c (c)=1$$
therefore:
$$log _a (c) + log _b (c)= log _{a+b} (c)$$
