Why is the series expansion for the logarithm not as 'popular,' or used as much as the series expansion for the exponential?

Question

This not a question for a class or homework; just asking for understanding -

The series expansion of the exponential seems to be widely used for a lot of things. It's used to define the number e, and in some cases, seems to be used as a definition of the exponential function itself. Most books I've read will at least include the series expansion for the exponential, if not use it for certain things.

My question is why are the series expansion(s) of ln(x) not as popular? Are they not as useful? Possibly not as well known? Or maybe lacking properties that the exponential series expansion has?

My apologies if I'm asking this question crudely.

Answer 1

One particular useful property of the exponential function is that its series expansion converges over all of $\mathbb{R}$. The logarithmic series expansion isn't nearly as nice. However, I often use the series expansion of $\ln(1+x)$, but that only converges for $|x| \leq 1$.

Answer 2

The exponential series is very general and it applies to all kind of algebraic objects; complex numbers, matrices, etc... And this has a lot of important applications; for example, the matrix exponential series solves any system of linear differential equations with constant coefficients.

Thus, the exponential function, and its series expansion, has a fundamental place in mathematics, which the logarithm has not. Of course this is not an absolute answer, just an opinion. This is unavoidable, since the original question is also subjective.