Reinventing the natural logarithm

The natural logarithm ln(x) can be defined as ∫(1 to x) 1/t dt. Using this definition, prove that it is a logarithm, i.e. as follows.

The logarithm of base g, log_g, is defined with the biimplication a=g^x ⇔ log_g(a)=x. Prove that there exists a constant e, such that log_e(a)=∫(1 to a) 1/t dt.

(Because this is a reinvention of the natural logarithm, be careful not to use any fact you know from the natural logarithm, including any fact of the constant e.)

(Proposed by Mauri Ericson Sombowadile/FB/Math)