What Is the Natural Logarithm?

The natural logarithm ( $e$ logarithm) $\ln (x)$ works the same way as the common logarithm when it comes to uses in equations. Like all logarithms, it follows the same rules as the rest.

Its appearance is also very similar. The only difference is that $\log (x)$ has 10 as the base number, which means $1 0^{\log x} = x$ , whilst $\ln (x)$ has $e$ as the base number, so $e^{\ln x} = x$ . The natural logarithm got its name because it has the natural number $e$ as the base.

Theory

The Natural Logarithm

The logarithm $\ln$ is a function. When you take the natural logarithm of a number ( $a$ ) you will get a new number $k$ . The number $k$

k = \ln (a), a > 0

is so that

\ln e^{x} = x and e^{\ln x} = x

In this figure you can see the graphs of the common logarithm and the natural logarithm. Both cross the $x$ -axis at $x = 1$ , but $\ln x$ grows slightly faster than $\log x$ .