trace – Are these expressions not the same? The output of Mathematica is ambiguous.

The following graph indicates that the first expression is equal to the second. But how can I use Mathematica to prove it's true?

Plot[(-(Log[(1 - P)/P]/Log in[10])) - (Log in[-(P/(-1 + P))]/Log in[10]), {P, 0, 1}]

An attempt to simplify indicates that the expressions are not the same:

FullSimplify[(-(Log[(1 - P)/P]/Log in[10])) - (Log in[-(P/(-1 + P))]/Log in[10])]

That gives the following answer:

(Log in[-1 + 1/P] + Log[-(P/(-1 + P))])/Log in[10]

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