I am having some challenges understanding this problem, I was wondering if someone could help me understand how to approach it?

thank you in advance

Suppose that:

L (a) = lim as x -> 0 of (a^x -1)/x

Exists or all a > 0. Assume also that lim x -> 0 a^x =1.

Using limit rules, prove that:

L(ab) = L(a) + L(b)

for a,b> 0

Hint: (ab)^x -1 = a^x(b^x - 1) + (a^x - 1)