What is the intuition for this logarithm law?

log*b**x* = (log*a**x*) / (log*a**b*)

There is intuition for other log laws. For example:

log(100) + log (1000)

= 2 factors of 10 + 3 factors of 10

= 5 factors of 10

= 5

= log (100x1000)

= (2 + 3) factors of 10

= 5 factors of 10

= 5

So more generally, log(*a*) + log(*b*) = log(*ab*). We're just counting the factors of 10 in *a* and *b*, so it doesn't matter whether we count them separately or together. It's equal either way.

Is there intuition for the equation in red above? Although it's trivial to prove, seeing the equation (and using it) in isolation seems more mechanical (e.g. plugging numbers into the Pythagorean Theorem because I see a right triangle) than logical (e.g. solving *x + 5 = 24* by subtracting 5 from both sides).