Does $\log_a b = \log_\sqrt a \sqrt b$ can be a basic logarithm law? Does the following equation is true for all $ a,b\in{\mathbb R}$?
$$\log_a b = \log_\sqrt a \sqrt b$$
I have tried to proove this, and I didnt find any contradiction. Is it true? 
EDIT
Thanks guys for the answers. I've just found a different proof without using $\ln$, for the record, here it is : 
$$\log_\sqrt{a} \sqrt{b} = \frac{ \log_a \sqrt{b}}{\log_a \sqrt{a}} = \frac{\frac{1}{2} \log_a b}{\frac{1}{2} \log_a a} = \frac{\log_a b}{1} = \log_a b$$
Second EDIT
Is it true to generelize this rule, like this?
$$\log_{a^n} {b^n} = \frac{ \log_a {b^n}}{\log_a {a^n}} = \frac{n \log_a b}{n \log_a a} = \frac{n\log_a b}{n} = \log_a b$$

It's true for all positive $b$, $a \neq 1$ as 
$$\log_\sqrt{a} \sqrt{b} = \frac{ \ln \sqrt{b}}{\ln \sqrt{a}} = \frac{\frac{1}{2} \ln b}{\frac{1}{2} \ln a} = \frac{\ln b}{\ln a} = \log_a b$$
