Logarithmus

Definition des Logarithmus zur Basis $b$

$\displaystyle x = \log_b a \Longleftrightarrow a = b^x \qquad b^{\log_{b}x}=x \qquad (e^{\ln x} = x)$

Logarithmus-Gesetze

$\displaystyle \log_x ( a \cdot b) = \log_x a + \log_x b$

$\displaystyle \log_x \left( \frac{a}{b} \right) = \log_x a - \log_x b$

$\displaystyle \log_x \left( a^b \right) = b \cdot \log_x a$

$\displaystyle \ln (a+b)= \ln a + \ln (1+\frac{b}{a})$

$\displaystyle \ln \sqrt[n]{a} = \ln a^{\frac{1}{n}} = \frac{1}{n} \ln a$

Basiswechsel

$\displaystyle \log_b a = c \Longleftrightarrow c = \frac{\log_x a}{\log_x b}$