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Problems PreCal 1508 PLTL Workshop, October 5, 2011\\
PL: Alex Knaust, Lecturer: Yi-Yu Liao\\

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Please do the problems that you feel will help your group the most first (you don't have to do them in order).All handouts are available at \url{http://alex.knaust.info/pltlfall2011/}
\begin{enumerate}
\item Determine the exact values of the following (without a calculator)
\begin{enumerate}
\item $\log_2{\sqrt[4]{8}}$
\item $\log_4{16^2}$
\item $\log_3{-3^{3}}$
\end{enumerate}

\item Rewrite the following expressions as a single logarithm if possible
\begin{enumerate}
\item $\ln{2} + \ln{x}$
\item $-4\log_6{2x}$
\item $3\ln{x} - 2\log_2{y}$
\item $\ln{x} - [\ln{(x+1)} + \ln{(x - 1)}]$
\item $\log_3{2x} + \frac{\log_5{y}}{\log_5{3}}$
\end{enumerate}

\item Solve for $x$ in the following equations
\begin{enumerate}
\item $e^{3x}  = 21$
\item $\left( 1 + \frac{0.10}{12}\right)^{12t} = 2$
\item $e^{2x} - 4e^x - 5 = 0\quad$ \emph{Hint : $e^{2x} = \left(e^{x}\right)^2$}
\item $\log 8x - \log{\left(1 + \sqrt{x}\right)}  = 2$
\item $\ln{x} + \ln{\left(x - 2\right)} = 1$
\end{enumerate}
\item Is the identity $z^{\log_x{y}} = y^{\log_x{z}}$ true? (Why/Why not?)
\item Is the identity $\left(\log_x{y}\right)^2 = \log_x{y^2}$ true? (Why/Why not?)
\item Given the following proof of $\log_b{xy} = \log_b{x} + \log_b{y}$ Can you use a similar technique to show the following identities?
\begin{proof}[Proof of $\log_b{xy} = \log_b{x} + \log_b{y}$]
\[xy = b^{\log_b{x}}\cdot b^{\log_b{y}} = b^{\log_b{x} + \log_b{y}}\]
\[\Longrightarrow \log_b{xy} = \log_b{b^{\log_b{x} + \log_b{y}}} = \log_b{x} + \log_b{y}\]
The first equations are a result of using the inverse property of logarithms, and the second line is a consequence of taking a logarithm of the first line.
\end{proof}
\begin{enumerate}
\item $\log_b{\frac{x}{y}} = \log_b{x} - \log_b{y}$
\item $\log_b{x^d} = d\cdot \log_b{x}$
\end{enumerate}
\end{enumerate}
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