Add so as to multiply
In a discussion of entropy it may be pointed out that when physical systems are paired the total number of microstates multiplies, since a microstate of the composite system is a pair of states of the separate systems. And yet the entropy, (which is a measure our ignorance of the exact state of a system) adds under these conditions. Boltzmann took this to mean that the entropy is proportional to the logarithm of the number of microstates, a famous result that endures today. This is because any logarithm satisfies the following functional equation, mapping multiplication of numbers to the addition of their logarithms.
But now we may ask which functions g actually satisfy this equation! If g is differentiable, then we may proceed as follows. First partial differentiate with respect to y.
Now set y to 1, and divide by x.
Look familiar? Integrating both sides gives a logarithm.
And we may write k for the coefficient of the logarithm, and note that if this is to satisfy the original functional equation, the constant of integration C must be zero. So the general differentiable solution is
But what if we care only about continuous functions: are there any more solutions? More on this soon!