# What is the difference between a $\log_{10}$ and logit transformation?

What is the difference between a $\log_{10}$ and logit transformation? I have tried to find the answer elsewhere but cannot find a strict distinction.

• It would be fine to refer to the logarithim function in general rather than specifically log10. Jul 28, 2015 at 21:19

The function $f(x)=\log_{10}(x)$ is the inverse of exponentiation with base 10. It is a monotonic injective function mapping positive numbers to $\mathbb{R}$. Positive numbers less than 1 are mapped to negative numbers. Positive numbers greater than one are mapped to positive numbers. In regression analysis, logarithm transformations are used when effects are multiplicative, especially when the regression involves something monetary like insurance claims or income. (The choice of base is not relevant to these properties; you could write the same about $\log_e$ and $\log_2.$)
The function $g(x)=\text{logit}(x)=\log\left(\frac{x}{1-x}\right)=\log(x)-\log(1-x)$ maps $x\in(0,1)$ to $\mathbb{R}$. It is also monotonic injective. The most common use for the logit function is that it can be used to transform probabilities from the unit interval to the real line; this plays a vital role in logistic regression, where one models the odds of two or more binary outcomes. The logit function also preserves symmetry about $0.5$. For example, $g(0.25)=-g(0.75).$
The logarithm function $f$ does not have the same role in linking real-valued numbers to probabilities. Likewise, the logarithm function does not have the same symmetry as the logit function. So those are two major differences.