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# Exponential and Logarithm 

##### Definition

Exponentiation $b^n$ corresponds to $n$ repeated multiplication of the base $b$:

$$b^{n} = \underbrace{b \times \dots \times b}_{n\,{\textrm {times}}}$$

with $b \in {\mathop{\mathbb R}}$ and $n \in {\mathop{\mathbb N}}$.

### Rules

For $n, m \in \Z$

 Sum $\displaystyle b^{m+n} = b^{m} \cdot b^{n}$ $\left( b^{m} \right)^{n} = b^{m \cdot n}$ Product $(b \cdot c)^{n} = b^{n} \cdot c^{n}$

## Exponential Function

$$\exp(x) \equiv e^x := \lim\limits_{n \rightarrow \infty} \left( 1 + \frac{x}{n} \right)^n = \sum\limits_{n = 0}^{\infty} \frac{x^n}{n!}$$

with the exponential function $\exp: {\mathop{\mathbb R}}{\rightarrow}{\mathop{\mathbb R}}$, the argument $x \in {\mathop{\mathbb R}}$, the Euler number $\mathrm{e} = 2,718281828..$

For complex numbers:

$${\mathrm{e}}^{{\boldsymbol{z}}} = {\mathrm{e}}^{a + b{\text{i}}} = {\mathrm{e}}^a \cdot ( \cos b + {\text{i}}\sin b )$$

with ${\boldsymbol{z}} \in {\mathop{\mathbb C}}$, $a, b \in {\mathop{\mathbb R}}$ and the imaginary unit ${\text{i}}$.

## Logarithm

The logarithm is the inverse function to exponentiation.

$$\log_{b}(x) = y \qquad \text{exactly if} \qquad b^{y} = x$$

with $b, x, y \in {\mathop{\mathbb R}}$

### Calculation Rules

For $b, x, y \in {\mathop{\mathbb R}}$

 Product $\displaystyle \log_{b}(x \cdot y) = \log_{b} x + \log_{b} y$ Quotient $\displaystyle \log_{b} \frac{x}{y} = \log_{b} x - \log_{b} y$ Power $\displaystyle \log_{b} \left( x^{p} \right) = p \cdot \log_{b}x$ Root $\displaystyle \log_{b} \sqrt[p]{x} = \frac{\log_b x}{p}$ Change base $\displaystyle \log_{b} x = \frac{\log_{k} x}{ \log_{k} b }$

Other notations: $\log_2 \equiv \lb$, $\log_{{\mathrm{e}}} \equiv \ln$, , $\log_{10} \equiv \lg$

### Taylor Series

$$\ln(z)={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots$$

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