\chapter{Exponentials and logarithms}
\section{Exponential equations}
Answer the questions.
\begin{enumerate}
\item
Explain the difference between the two exponential equations:
\[2^{x+1}=8\]
\[3^{x+1}=8\]
and the best method to solve them
\item
Suppose we want to solve the equation \(5^{2x+1}=9\), and we want to know a decimal approximation
of the answer with two decimal places. We consider two methods: the translation formula
for logarithms, or taking \(\ln\) of both sides. Explain why we cannot use the first method.
\item
To solve the equation \(3 e^{x+8}= 5\), we took \(\ln\) of both sides, and we found
\[\ln \left( 3 e^{x+8} \right) = \ln 5\]
What is the second step?
(i) Use property (C) of the logarithms: \(\ln x^y=y\ln x\)
(ii) Use the simplification formula \(\ln e^x = x\)
(iii) Use property (A) of the logarithms: \(\ln(xy) = \ln x + \ln y\).
\item
We are solving a certain exponential equation, and we reached the following equation:
\[3x \ln 4 -x \ln3 = 1+\ln 3.\]
What is the next step?
\end{enumerate}