# Are logarithmic functions one to one?

As a function from $\left(0 , \infty\right) \to \mathbb{R}$, logarithms are one to one.
Considering the natural logarithm, it is the inverse of the exponential function ${e}^{x} : \mathbb{R} \to \left(0 , \infty\right)$, which is strictly monotonically increasing, so $\ln : \left(0 , \infty\right) \to \mathbb{R}$ is itself strictly monotonically increasing and one to one.
Any other logarithm is expressible as a constant multiple of $\ln$, so is also one to one.