This question belongs in intermediate algebra or perhaps even pre-calculus, but it is an excellent question.

Logarithm functions, regardless of base, are members of a class of functions known as invertible. There is a theorem that proves

[tex]\text {If } f(x) \text { is invertible, then } f(x) = f(y) \implies x = y.[/tex]

It can be shown that logs are invertible (in my youth, when logs were actually used for computations, we had to work with the inverse called an antilog).

But if you do not want to work through the proof that logs are invertible, a proof that is very advanced, you can just say that one of the laws of logarithms is

[tex]log_a(x) = log_a(y) \implies x = y.[/tex]

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