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Definition

Exponentiation bnb^n corresponds to nn repeated multiplication of the base bb:

bn=b××bntimesb^{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 bm+n=bmbn\displaystyle b^{m+n} = b^{m} \cdot b^{n}
(bm)n=bmn\left( b^{m} \right)^{n} = b^{m \cdot n}
Product (bc)n=bncn(b \cdot c)^{n} = b^{n} \cdot c^{n}

Exponential Function

exp(x)ex:=limn(1+xn)n=n=0xnn!\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 $\exp: \R \rightarrow\R$, the argument $x \in \R$, the Euler number e=2,718281828..\mathrm{e} = 2,718281828..

For complex numbers:

e𝐳=ea+bi=ea(cosb+isinb)\mathrm{e}^{\boldsymbol{z}} = \mathrm{e}^{a + b\text{i}} = \mathrm{e}^a \cdot ( \cos b + \text{i}\sin b )

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

Logarithm

The logarithm is the inverse function to exponentiation.

logb(x)=yexactly ifby=x\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 logb(xy)=logbx+logby\displaystyle \log_{b}(x \cdot y) = \log_{b} x + \log_{b} y
Quotient logbxy=logbxlogby\displaystyle \log_{b} \frac{x}{y} = \log_{b} x - \log_{b} y
Power logb(xp)=plogbx\displaystyle \log_{b} \left( x^{p} \right) = p \cdot \log_{b}x
Root logbxp=logbxp\displaystyle \log_{b} \sqrt[p]{x} = \frac{\log_b x}{p}
Change base logbx=logkxlogkb\displaystyle \log_{b} x = \frac{\log_{k} x}{ \log_{k} b }

Other notations: $\log_2 \equiv \lb$, logeln\log_{\mathrm{e}} \equiv \ln, , log10lg\log_{10} \equiv \lg

Taylor Series

ln(z)=(z1)11(z1)22+(z1)33(z1)44+\ln(z)={\frac {(z-1)^{1}}{1}}-{\frac {(z-1)^{2}}{2}}+{\frac {(z-1)^{3}}{3}}-{\frac {(z-1)^{4}}{4}}+\cdots