basic rules logarithm of exponential

Question

I am looking for proof of the basic rules of logarithm. I can prove all basic rules except this $$\log_ab^y=y\log_ab$$ how to get this rule using definition of logarithm.

Answer 1

If $\log_ab=m\iff a^m=b$

So, $\log_ab^y=n\iff a^n=b^y=(a^m)^y=a^{my}\implies n=my$ if $a\ne0,1$

So, $\log_ab^y=n=my=y\log_ab$

Answer 2

Just check that $$ a^{y \log_{a}(b)} = a^{\log_{a}(b) y} = \left(a^{\log_{a}(b)}\right)^{y} = b^{y}. $$

Answer 3

Assuming you know, or have proven that $\log_a(xy) = \log_a x + \log_z y$ We use this to prove:

$$\log_ab^y= \log_a \underbrace{(b \cdot b \cdot ...\cdot b)}_{\large y \; times}$$ $$\iff \log_a b^y = \underbrace{\log_a b + \log_a b + \cdots + \log_z b}_{\large y \;\;times}$$ $$ \iff \log_ab^y = y\log_z b$$

Else: by definition only

$$a^{y \log_{a}(b)}= a^{\log_{a}(b) y}=\left(a^{\log_{a}(b)}\right)^{y}=b^{y}.$$