Why doesn't the binary classification log loss formula make it explicit that natural log is being used?
This comes down to the change-of-base formula. For any two numbers $a$ and $b$, the following equation is true.

$$
\log_a(x) = \frac{\log_b(x)}{\log_b(a)}.
$$

What this means is that the errors are proportional. So if you wanted to change to using $\log_{10}$, you would simply end up multiplying by a constant factor, and model selection would be the same.

Explicitly,

$$logloss(N=1) = \frac{y \log_{10}(p) + (1 - y) \log_{10}(p)}{\log_{10}(e)}$$

Or, equivalently

$$
logloss(N=1) \cdot \log_{10}(e) = y \log_{10}(p) + (1 - y) \log_{10}(p)
$$



In other words: The base of the logarithm doesn't matter because everything ends up being proportional.