Dividing logs with same base

Question

Problem

$\dfrac{\log125}{\log25} = 1.5$

From my understanding, if two logs have the same base in a division, then the constants can simply be divided i.e $125/25 = 5$ to result in ${\log5} = 1.5$ but that is not the case as ${\log5} \neq 1.5$ .

Correct answer

Each log can be rewritten to be $\frac{3\log5}{2\log5} = 1.5$ therefore $\frac{3}{2} = 1.5$

I'm unsure why this is correct over the previous method.

My question

What was wrong with simply dividing the constants $125/25 = 5$ versus rearranging the logarithm?

Answer 1

Your "understanding" is just totally wrong. It's not true that $\frac{\log a}{\log b}=\log(a/b)$ in general, and indeed this problem is a counterexample.

Answer 2

Dividing logs which have the same base changes the base of the log.

That is $\frac {\log a}{\log b} = \log_b a$

It doesn't matter what base we were using on the left hand side. It will change the base of the log as above.

$\frac {\log 125}{\log 25} = \log_{25} 125$ and $25^{\frac 32} = 125$