Definition:Natural Logarithm

Positive Real Numbers
Let $x \in \R$ be a real number such that $x > 0$.

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


 * $\displaystyle \ln x := \int_1^x \frac {\mathrm dt} t$

For any definition of the exponential function that isn't "the function inverse of the natural logarithmic function", the natural logarithmic function can be defined as the function inverse of the exponential function.

Complex Numbers
The complex natural logarithm of a complex value $z \in \C$ is written $\log \left({z}\right)$ (no base value) and is defined:


 * $\log \left({z}\right) := \ln \left|{z}\right| + i \arg \left({z}\right)$.

The principal branch of the complex logarithm is written and defined:


 * $\operatorname{Log} \left({z}\right) := \ln \left|{z}\right| + i \operatorname{Arg} \left({z}\right)$.

where $\arg \left({z}\right)$ is the continuous argument of $z$ and $\operatorname{Arg}\left({z}\right) = \arg \left({z}\right) \ \left({\bmod \left({2 \pi}\right)}\right)$.

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


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

The first of these is fairly commonly encountered, and frequently preferred. The second is ambiguous (it doesn't tell you which base it is the logarithm of) and the third is verbose.

However, notation is misleadingly inconsistent throughout the literature, and if there is any confusion about exactly what is meant, the full (verbose) format can be argued for.