Properties-of-logarithms-for-real-numbers.tex

\Section{Properties of logarithms for real numbers}

\begin{multicols}{2}

 \begin{FormulaBox}{Domain and range}
  Domain of $\me^x$ is all real numbers

  But $\me^x > 0$ for every $x$
  \smallskip

  Domain of $\LN{x}$ is only $x > 0$

  But $\LN{x}$ is
  \begin{itemize}
  \item positive for $x > 1$
  \item zero for $x = 1$
  \item negative for $0 < x < 1$
  \end{itemize}

 \end{FormulaBox}
 \begin{FormulaBox}{Special values}
  \begin{equation*}
   \begin{aligned}
     \me^0 &=1  &\quad \LN{1} &= 0
     \\
     \me^1 &=\me &\quad \LN{\me} &=1
   \end{aligned}
  \end{equation*}
 \end{FormulaBox}
 \begin{FormulaBox}{Inverse function}
  \FormulaCompact{\me^{\LN{\Y}}}{\Y}
  \FormulaCompact{\LN{\me^{\Y}}}{\Y}
 \end{FormulaBox}
 \begin{FormulaBox}{Multiplication / Addition}
  \FormulaCompact{\me^{\THIS} \cdot \me^{\THAT}}{\me^{\THIS + \THAT}}
  \FormulaCompact{\LN{\U \cdot \V}}{\LN{\U} + \LN{\V}}
 \end{FormulaBox}
 \begin{FormulaBox}{Division / Subtraction}
  \FormulaCompact{\frac{\me^{\THIS}}{\me^{\THAT}}}{\me^{\THIS - \THAT}}
  \FormulaCompact{\LN{\frac{\Hi}{\Lo}}}{\LN{\Hi} - \LN{\Lo}}
 \end{FormulaBox}
 \begin{FormulaBox}{Nested power / Multiplication}
  \FormulaCompact{\left(\me^\N\right)^\M}{\me^{\N\cdot\M}}
  \FormulaCompact{\LN{\B^\P}}{\P\cdot\LN{\B}}
 \end{FormulaBox}
 \begin{FormulaBox}{Using $\ln$ and $\me$ to compute powers}
  \WithSymbolDefs{
   \begin{equation*}
    \B^\P = \me^{\P \cdot \LN{\B}}
   \end{equation*}
  }
 \end{FormulaBox}
 \begin{FormulaBox}{Base other than $\me$}
  \WithSymbolDefs{
   \begin{equation*}
    \P = \log_{\B}\left(\Y\right)
    \text{ means }
    \B^\P = \Y
   \end{equation*}}
  Conversion to $\ln$:
  \WithSymbolDefs{
   \begin{equation*}
    \log_{\B}\left(\Y\right)
    = \frac{\LN{\Y}}{\LN{\B}}
   \end{equation*}
  }
 \end{FormulaBox}
 \begin{FormulaBox}{Calculator keys}
  On most calculators
  \begin{itemize}
  \item \texttt{EXP} means $\me^x$
  \item \texttt{LN} means $\LN{x}$
  \item \texttt{LOG} means $\log_{10}(x)$
  \end{itemize}
 \end{FormulaBox}
\end{multicols}

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