Exponential and Logarithm#

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

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

with $$b \in \R$$ and $$n \in \N$$.

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 $$\exp: \R \ra \R$$, the argument $$x \in \R$$, the Euler number $$\mathrm{e} = 2,718281828..$$

For complex numbers: $$\(\e^{\cx z} = \e^{a + b\i} = \e^a \cdot ( \cos b + \i \sin b )$$\) with $$\cx z \in \C$$, $$a, b \in \R$$ and the imaginary unit $$\i$$.

Logarithm#

The logarithm is the inverse function to exponentiation.

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

with $$b, x, y \in \R$$

Calculation Rules#

For $$b, x, y \in \R$$

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

Taylor Series#

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