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

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