Kunnskapsgnist
$a$-logaritmen av et tall $x$ er lik det tallet som $a$ må opphøyes i for å få $x$:
$$\log_{\textcolor{red}{a}}(\textcolor{blue}{x}) = \textcolor{green}{y} \qquad \Rightarrow \qquad \textcolor{red}{a}^{\textcolor{green}{y}} = \textcolor{blue}{x}$$Noen nyttige logaritmeregler:
$$\begin{aligned} \textnormal{Fordi } \textcolor{red}{a}^{\textcolor{green}{0}} = \textcolor{blue}{1}: \qquad & \log_{\textcolor{red}{a}} (\textcolor{blue}{1}) = \textcolor{green}{0} \\ \textnormal{Fordi }\textcolor{red}{a}^{\textcolor{green}{1}} = \textcolor{blue}{a}: \qquad & \log_{\textcolor{red}{a}} (\textcolor{blue}{a}) = \textcolor{green}{1} \\ \textnormal{Multiplisering: } \qquad & \log_{\textcolor{red}{a}} (\textcolor{blue}{x} \cdot \textcolor{blue}{y})= \log_{\textcolor{red}{a}}(\textcolor{blue}{x}) + \log_{\textcolor{red}{a}}(\textcolor{blue}{y}) \\ \textnormal{Dividering: } \qquad & \log_{\textcolor{red}{a}} \left( \frac{\textcolor{blue}{x}}{\textcolor{blue}{y}} \right)= \log_{\textcolor{red}{a}} (\textcolor{blue}{x}) - \log_{\textcolor{red}{a}} (\textcolor{blue}{y}) \\ \textnormal{Eksponenter: } \qquad & \log_{\textcolor{red}{a}} (\textcolor{blue}{x}^{\textcolor{green}{y}}) = \textcolor{green}{y} \log_{\textcolor{red}{a}} (\textcolor{blue}{x}) \\ \textnormal{Skifte grunntall: } \qquad & \log_{\textcolor{red}{a}}(\textcolor{blue}{x}) = \frac{\log_{\textcolor{green}{b}}(\textcolor{blue}{x})}{\log_{\textcolor{green}{b}}(\textcolor{red}{a})} \end{aligned}$$der $x > 0$ og $y > 0$.
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