The logarithm as a measure of precision

 This is the standard property of the logarithm. But this property relies on uniform convergence of the integrals as dx or dt go to zero.  The integrand, 1/x, is really the precision, the spacing of the number line relative to the size of the number x. If the number line retains accuracy as x gets larger then the number line must increase in precision, or decrease spacing.  in the limit, as dx goes to zero, the precision of the number line must go to perfect accuracy, spacing between numbers goes to zero.  The accuracy of the number line increases, monotonically as x gets larger. At infinity, there is a one to one mapping between x and the number line, so precision must add.  The log just gives you the precision of the number line at x. The proof above is just a tautology since the integrals will not converge uniformly unless the number line obeys logarithmic precision.

The grammar involving exponents is simply a standard to map precision to size.

The more general definition of the logarithm is:

This computes the rate of change in the size of f relative to f.