# Logarithm Identity

I am reviewing algebra 2 using some video tutorial from mathtutordvd.com.

In one of the videos, the author converts a logarithm equation into the exponent form as follows:

Which logarithm identity is used to convert the equation?

• That's just because the exponential function is well defined. – mwt Sep 26 '18 at 18:58
• They just took the exponential of both sides... if $x=y$ then $6^x=6^y$ – user438666 Sep 26 '18 at 19:03
• This being said, it is far too complicated. Logarithms in any base are bijections from $\mathbf R_+$ to $\mathbf R$, so $\log_n(2x-3)=\log_b 4$ implies $2x-3=4$. – Bernard Sep 26 '18 at 19:03

$$\log_a b=c\iff a^c=a^{\log_a b}=b$$
but since $$\log$$ function is injective we can conclude directly
$$\log_a f(x)=\log_a g(x)\iff f(x)=g(x)$$
for $$f(x),g(x)>0$$.