\begin{frame} \frametitle{Natural Logarithm} \begin{block}{} The \emph{natural logarithm} $\ln$ is a special logarithm with base $e$: \begin{talign} \ln x = \log_e x \end{talign} \end{block} \pause \begin{exampleblock}{} Solve the equation $e^{5-3x} = 10$.\pause \begin{talign} \ln (e^{5-3x}) &= \ln 10 &&\text{apply natural logarithm on both sides}\\ \mpause[1]{5-3x} &\mpause[1]{= \ln 10} \\ \mpause[2]{3x} &\mpause[2]{= 5 - \ln 10} \\ \mpause[3]{x} &\mpause[3]{= \frac{5 - \ln 10}{3}} \end{talign} \end{exampleblock} \pause\pause\pause\pause \begin{exampleblock}{} Express $\ln a + \frac{1}{2} \ln b$ in a single logarithm.\pause \begin{talign} \ln a + \frac{1}{2} \ln b &\mpause[1]{= \ln a + \ln b^{\frac{1}{2}}} \mpause[2]{= \ln a + \ln \sqrt{b}} \mpause[3]{= \ln (a\sqrt{b})} \end{talign} \end{exampleblock} \end{frame}