# What is the difference between common and natural logarithms?

Aug 16, 2015

The base.

Essentially the common log is ${\log}_{10}$, the inverse of ${10}^{x}$, while the natural log is ${\log}_{e}$, the inverse of ${e}^{x}$.

#### Explanation:

The common logarithm is useful for base $10$ calculations, especially in conjunction with scientific notation.

The natural logarithm ${\log}_{e} x = \ln x$ is used more in algebra and calculus. Its inverse ${e}^{x}$ has nice properties like $\frac{d}{\mathrm{dx}} {e}^{x} = {e}^{x}$, ${e}^{i \theta} = \cos \theta + i \sin \theta$, etc.

Another frequently used base for logarithms is $2$. The binary logarithm ${\log}_{2}$ is often used in computer science.

It is easy to convert between different logarithmic bases using the change of base formula:

${\log}_{a} b = \frac{{\log}_{c} b}{{\log}_{c} a}$

For example ${\log}_{10} x = \frac{\ln x}{\ln 10}$ and $\ln x = \ln 10 \cdot {\log}_{10} x$