# What are the differences between using the natural log versus base-10 log for the logistic regression?

Normally logistic regression uses the natural logarithm, as opposed to a $$\log_{10}$$ base. If we have a regression for binary $$Y$$ where $$p = \Pr(Y=1\mid X)$$,

$$\log_{10}\left(\dfrac{p}{1-p}\right) = \beta_0 + \beta_1X_1$$

and we found $$\widehat{\beta_0} = -2$$ and $$\widehat{\beta_1} = 3$$, then we can interpret $$\widehat{\beta_0}$$ as saying the log odds of $$Y=1$$ are 100 to 1 when $$X_1= 0$$. Similarly, $$\widehat{\beta_1}$$ can be interpreted by saying that if $$X_1$$ increasing by 1, it implies that the odds of $$Y=1$$ increases by $$10^3$$.

Why isn't this interpretation preferred to what you would get from the natural log $$\ln$$?

As you may know, logistic regression is fit via maximum likelihood. We often instead maximize the log likelihood because it is easier and prevents underflow when using a computer. We use the natural log for the log likelihood because we have to take derivatives of the log likelihood for maximization. The natural log is nicer because we don't need to book keep the extra factor of $$1/\ln(a)$$ in the derivative when using a different base.