Definition:Logarithm

Natural Logarithm
Let $$x \in \mathbb{R}$$ be a real number such that $$x > 0$$.

The (natural) logarithm of $$x$$ is defined as:

$$\mathbf {Define:} \ \ln x \ \stackrel {\mathbf {def}} {=\!=} \ \int_1^x \frac {dt} t$$

Notation
The natural logarithm of $$x$$ is written variously as:


 * $$\ln x$$
 * $$\log x$$
 * $$\log_e x$$

The first of these is the most common and generally prefered. The second is ambiguous (it doesn't tell you which base it is the logarithm of) and the third is verbose.