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\centerline{MATH 135 -- Calculus 1}
\centerline{Logarithm Functions}
\centerline{September 14, 2016}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
As my high school math teacher Mr. Brennan said, {\it ``a logarithm is an exponent}.''  He
was a \emph{total nerd} (maybe you would say I take after him and I am too!).  But he was right!  
Of course, this is another way of saying the key property defining the $\log_b$ function:
$$y = \log_b x \Longleftrightarrow x = b^y,$$
so $\log_b(b^y) = y$ for all real $y$, and $b^{\log_b(x)} = x$ for all $x > 0$.  The most important
logarithm function for us is the \emph{natural logarithm} $\ln(x)$, which is the same as $\log_e(x)$ where
$e \doteq 2.71828\cdots$ is (for now) a somewhat mysterious number(!)  From the properties of exponents,
(see the table on page 41 of our text), we get the key properties of the $\ln$ function:
\begin{enumerate}
\item[(1)]  $\ln(x_1 x_2) = \ln(x_1) + \ln(x_2)$,
\item[(2)]  $\ln\left(\frac{x_1}{x_2}\right) = \ln(x_1) - \ln(x_2)$, and 
\item[(3)]  $\ln\left(A^B\right) = B\ln(A)$,
\end{enumerate}
and similarly for $\ln_b$ for any $b > 0$.  
Today, we want to practice using these properties of the logarithm functions to simplify expressions
and solve equations. 

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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)]  Simplify and compute the value. (Do these \emph{without using a calculator}!)
\begin{enumerate}
\item[(a)]  $\log_2\left(\frac{1}{128}\right)$
\item[(b)]  $\log_5(125)$
\item[(c)]  $\ln(e^4) + \ln\left(\frac{1}{e^5}\right)$
\end{enumerate}
\item[2)]  Solve each of these equations for the unknown by using properties of logarithms.
Get an exact answer (expressed using logarithms), then find a decimal approximation using a calculator.
\begin{enumerate}
\item[(a)]  $e^{5x + 1} = 2$.
\item[(b)]  $2^{2x} = 3^{3x-5}$.
\item[(c)]  $e^{2t + 1} = 7^{t - 4}$  
\end{enumerate}
\item[3)]  
\begin{enumerate}
\item[(a)]  Explain why $f(x) = e^{2(x-1)} + 1$ has an inverse function.  (Hint:  What is the property
that says a function has an inverse function?  Sketch the graph $y = e^{2(x-1)} + 1$ to see that this
function ``has it.'')
\item[(b)]  Find a formula for $f^{-1}$ for the function from part (a).  Sketch the graph $y = f^{-1}(x)$
on the same axes as you had in part (a).  
\end{enumerate}
\end{enumerate}
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