Generally, Log(a^b) is b*Log(a). The log of a larger number will be greater than the log of a smaller number. Taking Logarithm allows comparing larger numbers.

Translation of the expression A^B to log(A^B) is just scaling the expression while maintaining the proportion. So the comparison less than or greater than still works.

Remember that the logarithm is like asking for a number that need to be the power of other number to be equal to something: log(a^b) = y

SUM PROPERTY
You need to first need to understand: log(A*B) = log(A) + log(B)

Let's say:
--> X = logb(M)
--> Y = logb(N)

Then:
b^X = M
b^Y = N

Then:
M*N = b^X * b^Y = b^(X+Y)

Logarithm in both sides:
logb(M*N) = logb(b^(X+Y)) // Here both bases cancel each other. Example: log10(10) == 1
So we ended up with: logb(M*N) = X + Y

Then just rewrite X and Y: logb(M*N) = logb(M) + logb(N)

The next steps are easy:

log(A*A) = log(A) + log(A) = log(A^2)

A^3 = A*A*A
log(A*A*A) = log(A) + log(A) + log(A) = log(A^3) = 3 * log(A)

So log(A^B) = B * log(A)