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I was trying to define a logarithmic function abstractly as a map from R

^{+}to R satisfying those 2 conditions that this question came to my mind. It's obvious that there are infinitely many functions satisfying the 1st condition (each logarithmic function in an arbitrary base satisfies the first condition). but I hope that the 2nd condition could be used to prove that only one fixed function satisfies both conditions (namely, I want to prove that only log

_{a}satisfies both conditions and then I'll use this idea to abstractly define the map log

_{a}: R

^{+}-> R).

Is there any way to prove or disprove this conjecture? I've been trying to prove it from tomorrow morning and have failed so far. my idea is to define h(x) = f(x) - g(x) where f and g are functions with the same domain satisfying those 2 conditions and then to show that h(x)=0 for all x's in the domain.