In this section, we’ll now combine what we know about exponential and logarithmic functions, as well as the properties of logarithms, in order to solve exponential and logarithmic equations.
Rewrite each logarithmic statement as an exponential statement.
\(\displaystyle \log_3(243) = 5\)
\(\displaystyle \log_{16}(4) = \frac12\)
(b)
Solve \(\log_5(x) = 3\)
(c)
If \(f(x) = x+8\text{,}\) what is the inverse function of \(f\text{?}\)
(d)
If \(f(x) = 5x\text{,}\) what is the inverse function of \(f\text{?}\)
(e)
How do inverse operations or inverse functions help us solve equations?
ExercisesPractice Exercises
1.
Solve each of the following algebraically. Use a calculator to approximate any irrational solutions.
(a)
\(3^{5x-6} = 81\)
(b)
\(9^{x-11} = 7\)
(c)
\(e^{x+3} + 4 = 19\)
(d)
\(5^{x-6} = 3^{2x+7}\)
2.
Solve each of the following algebraically. Be sure to confirm any solutions are not extraneous.
(a)
\(\log_6(3x+1) = \log_6(x-9)\)
(b)
\(\log_7(x-4)+3 = 5\)
(c)
\(\log_3(x-2) = 1 - \log_3(x-4)\)
SubsectionDefinitions
Definition2.4.1.Logarithm.
For any real number \(x \gt 0\text{,}\) the logarithm with base \(b\) of \(x\), where \(b \gt 0\) and \(b \neq 1\text{,}\) is denoted by \(\log_b(x)\) and is defined by
\begin{equation*}
y = \log_b(x) \text{ if and only if } x = b^y
\end{equation*}
Definition2.4.2.One-to-One Property of Exponential Functions.
For any algebraic expressions \(S\) and \(T\text{,}\) and any positive number \(b\text{,}\) with \(b \neq 1\text{,}\)
\begin{equation*}
b^S = b^T \text{ if and only if } S = T
\end{equation*}
Definition2.4.3.One-to-One Property of Logarithmic Functions.
For any algebraic expressions \(S \gt 0\) and \(T \gt 0\text{,}\) and any positive number \(b\text{,}\) with \(b \neq 1\text{,}\)
\begin{equation*}
\log_b(S) = \log_b(T) \text{ if and only if } S = T
\end{equation*}
Note: Because \(\log_b(x)\) has the domain \((0,\infty)\) for all \(b \gt 0, b \neq 1\text{,}\) when we solve an equation involving logarithms, we must always check to see if the solution we’ve found is valid or if it is an extraneous solution.
ExercisesExit Exercises
1.
(a)
What’s the general process for solving exponential equations that have one exponential expression in them?
(b)
Why can logarithmic equations have extraneous solutions and how can an extraneous solution be recognized?
(c)
Solve \(4e^{2k+1}+3 = 27\)
(d)
Solve \(\log_8(5x+12) - \log_8(x) = \log_8(2)\)
ReflectionReflection
1.
On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?
