How ought a Logarithm to be defined?

Extract

Mr. Fletcher begins his article in the January number of the Gazette by giving two reasons why a logarithm ought not to be defined as the index of the power of the base which is equal to the number. But are these reasons well-founded? One is that it does not afford a basis for the more advanced parts of the theory of logarithms. That statement would require a proof that might be very difficult or perhaps impossible to produce.