We all know the series expansion
$$\log 2=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}n. \tag1$$
I also am able to use the method of [Wilf-Zeilberger][1] to the effect that
$$\log 2=3\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n\binom{2n}n2^n}.  \tag2$$

>**QUESTION.** Can you provide yet another proof of the formula in (2)?

**Remark.** My motivation for this question goes beyond this particular series, hoping it paves a way forward in my study.

[1]: https://en.wikipedia.org/wiki/Wilf%E2%80%93Zeilberger_pair