Definition:Logarithm

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

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

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

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

$$\log (z) \ \stackrel {\mathbf {def}} {=\!=} \ $$ $$ \ln |z| + i \text{ arg} (z) \ $$.

The principal branch of the complex logarithm is written and defined

$$\text{Log} (z) \ \stackrel {\mathbf {def}} {=\!=} \ $$ $$\ln|z| + i\text{ Arg}(z) \ $$.

where $$\text{ arg}(z) \ $$ is the continuous argument of $$z \ $$ and $$\text{ Arg}(z) = \text{ arg}(z) \ (\text{mod } 2\pi) \ $$.

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.