$f(x) = \sum_{n=1}^{\infty}(-1)^n \frac{x^{2n-1}}{(2n)! \ln 2n}$ is totally monotone?

Question

I think

$f(x) = \sum_{n=1}^{\infty}(-1)^n \frac{x^{2n-1}}{(2n)! \ln 2n}$ is totally monotone.

Bernstein's theorem and his integral transform convinced me.

Am I correct ?

Probably related or useful is a function like

$f(x,k) = \sum_{n=1}^{\infty}(-1)^n \frac{x^{2n-1}}{(2n)! \ln (2n+k)}$

which follows or almost follows from differentiating.

This is a follow-up of the question

A curious limit for $-\frac{\pi}{2}$

about the same function.