Objectives
- Convert between exponential and logarithmic form. Evaluate a logarithmic function, including common and natural logarithms.
Section 5.3 Introduction to Logarithms (EL3)
Subsection 5.3.1 Activities
In Section 2.6, we introduced the idea of an inverse function. The fundamental idea is that \(f\) has an inverse function if and only if there exists another function \(g\) such that \(f\) and \(g\) “undo” one another’s respective processes. In other words, the process of the function \(f\) is reversible, and reversing \(f\) generates a related function \(g\text{.}\)
Activity 5.3.1.
Let \(P(t)\) be the function given by \(P(t)=10^t\text{.}\)
(a)
Fill in the table of values for \(P(t)\text{.}\)
| \(t\) | \(y=P(t)\) |
|---|---|
| \(-3\) | |
| \(-2\) | |
| \(-1\) | |
| \(0\) | |
| \(1\) | |
| \(2\) | |
| \(3\) |
Answer.
| \(t\) | \(y=P(t)\) |
|---|---|
| \(-3\) | \(0.001=10^{-3}\) |
| \(-2\) | \(0.01=10^{-2}\) |
| \(-1\) | \(0.1=10^{-1}\) |
| \(0\) | \(1=10^{0}\) |
| \(1\) | \(10=10^{1}\) |
| \(2\) | \(100=10^{2}\) |
| \(3\) | \(1000=10^{3}\) |
(b)
Do you think P will have an inverse function? Why or why not?
Answer.
Students may say, each input has a distinct output, the function is one-to-one.
(c)
Since \(P\) has an inverse function, we know there exists some other function, say \(L\text{,}\) such that \(y=P(t)\) represent the same relationship between \(t\) and \(y\) as \(t=L(y)\text{.}\) In words, this means that \(L\) reverses the process of raising to the power of 10, telling us the power to which we need to raise 10 to produce a desired result. Fill in the table of values for \(L(y)\text{.}\)
| \(y\) | \(L(y)\) |
|---|---|
| \(10^{-3}\) | |
| \(10^{-2}\) | |
| \(10^{-1}\) | |
| \(10^{0}\) | |
| \(10^{1}\) | |
| \(10^{2}\) | |
| \(10^{3}\) |
Answer.
| \(y\) | \(L(y)\) |
|---|---|
| \(10^{-3}\) | \(-3\) |
| \(10^{-2}\) | \(-2\) |
| \(10^{-1}\) | \(-1\) |
| \(10^{0}\) | \(0\) |
| \(10^{1}\) | \(1\) |
| \(10^{2}\) | \(2\) |
| \(10^{3}\) | \(3\) |
(d)
What are the domain and range of \(P\text{?}\)
Answer.
Domain \((-\infty, \infty)\) Range \((0,\infty)\)
(e)
What are the domain and range of \(L\text{?}\)
Answer.
Domain \((0,\infty)\) Range \((-\infty, \infty)\)
Remark 5.3.2.
The powers of 10 function \(P(t)\) has an inverse \(L\text{.}\) This new function \(L\) is called the base 10 logarithm. But, we could have done a similar procedure with any base, which leads to the following definition.Definition 5.3.3.
The base \(b\) logarithm of a number is the exponent we must raise \(b\) to get that number. We represent this function as \(y=\log_b(x)\text{.}\)
We read the logarithmic expression as "The logarithm with base \(b\) of \(x\) is equal to \(y\text{,}\)" or "log base \(b\) of \(x\) is \(y\text{.}\)"
Remark 5.3.4.
We can use Definition 5.3.3 to express the relationship between logarithmic form and exponential form as follows:
\begin{equation*}
\log_{b}(x)=y \iff b^{y}=x
\end{equation*}
whenever \(b \gt 0, b \neq 1 \)Activity 5.3.5.
Write the following logarithmic equations in exponential form.
(a)
\(\log_{7} (\sqrt{7})=\dfrac{1}{2}\)
- \(\displaystyle 7^{\frac{1}{2}}=\sqrt{7}\)
- \(\displaystyle 7^{\sqrt{7}}=\dfrac{1}{2}\)
- \(\displaystyle \sqrt{7}^{\frac{1}{2}}=7\)
- \(\displaystyle \frac{1}{2}^{7}=\sqrt{7}\)
Answer.
A
(b)
\(\log_{3} (m)=r\)
- \(\displaystyle 3^{m}=r\)
- \(\displaystyle r^{3}=m\)
- \(\displaystyle 3^{r}=m\)
- \(\displaystyle m^{r}=3\)
Answer.
C
(c)
\(\log_{2} (x)=6\)
- \(\displaystyle 2^{x}=6\)
- \(\displaystyle 6^{2}=x\)
- \(\displaystyle x^{x}=6\)
- \(\displaystyle 2^{6}=x\)
Answer.
D
Activity 5.3.6.
Write the following exponential equations in logarithmic form.
(a)
\(5^{2}=25\)
- \(\displaystyle \log_{5} (2)=25\)
- \(\displaystyle \log_{5} (25)=2\)
- \(\displaystyle \log_{25} (5)=2\)
- \(\displaystyle \log_{2} (25)=5\)
Answer.
B
(b)
\(3^{-1}=\frac{1}{3}\)
- \(\displaystyle \log_{3} (-1)=\frac{1}{3}\)
- \(\displaystyle \log_{\frac{1}{3}} (-1)=3\)
- \(\displaystyle \log_{3} \left(\frac{1}{3}\right)=-1\)
- \(\displaystyle \log_{-1} \left(\frac{1}{3}\right)=3\)
Answer.
C
(c)
\(10^{a}=n\)
- \(\displaystyle \log_{10} (n)=a\)
- \(\displaystyle \log_{10} (a)=n\)
- \(\displaystyle \log_{n} (10)=a\)
- \(\displaystyle \log_{a} (n)=10\)
Answer.
A
Activity 5.3.7.
We can use the idea of converting a logarithm to an exponential to evaluate logarithms.
(a)
Consider the logarithm \(\log_{3}(9)\text{.}\) If we want to evaluate this, which question should you try and solve?
- To what exponent must \(9\) be raised in order to get \(3\text{?}\)
- What exponent must be raised to the third in order to get \(9\text{?}\)
- To what exponent must \(3\) be raised in order to get \(9\text{?}\)
- What exponent must be raised to the ninth in order to get \(3\text{?}\)
Answer.
C
(b)
Evaluate the logarithm, \(\log_{3}(9)\text{,}\) by answering the question from part (a).
Answer.
\(2\)
Activity 5.3.8.
Evaluate the following logarithms.
(a)
\(\log_2(8)\)
- \(\displaystyle 4\)
- \(\displaystyle \frac{1}{4}\)
- \(\displaystyle -3\)
- \(\displaystyle 3\)
Answer.
D
(b)
\(\log_{144}(12)\)
- \(\displaystyle \frac{1}{2}\)
- \(\displaystyle -2\)
- \(\displaystyle 2\)
- \(\displaystyle -\frac{1}{2}\)
Answer.
A
(c)
\(\log_{10}\left(\frac{1}{1000}\right)\)
- \(\displaystyle \frac{1}{3}\)
- \(\displaystyle -3\)
- \(\displaystyle 3\)
- \(\displaystyle -\frac{1}{3}\)
Answer.
B
(d)
\(\log_{e}\left(e^{3}\right)\)
- \(\displaystyle 3\)
- \(\displaystyle e^{3}\)
- \(\displaystyle -3\)
- \(\displaystyle \frac{1}{3}\)
Answer.
A
(e)
\(\log_{7}(1)\)
- \(\displaystyle 7\)
- \(\displaystyle \frac{1}{7}\)
- \(\displaystyle 0\)
- \(\displaystyle 1\)
Answer.
C
Remark 5.3.9.
Consider the results of Activity 5.3.8 part (d) and (e). Using the rules of exponents and the fact that exponents and logarithms are inverses, these properties hold for any base:
\begin{equation*}
\log_{b}(b^{x})=x
\end{equation*}
\begin{equation*}
\log_b(1)=0
\end{equation*}
Remark 5.3.10.
There are some logarithms that occur so often, we sometimes write them without noting the base. They are the common logarithm and the natural logarithm.
The common logarithm is a logarithm with base \(10\) and is written without a base.
\begin{equation*}
\log_{10}(x)=\log (x)
\end{equation*}
The natural logarithm is a logarithm with base \(e\) and has its own notation.
\begin{equation*}
\log_{e}(x)=\ln (x)
\end{equation*}
Activity 5.3.11.
Evaluate the following logarithms. Some may be done by inspection and others may require a calculator.
(a)
\(\log_4 \left( \frac{1}{64}\right)\)
Answer.
\(-3\)
(b)
\(\ln \left( 1\right)\)
Answer.
\(0\)
(c)
\(\ln \left( 12 \right)\)
Answer.
\(2.485\)
(d)
\(\log \left( 100\right)\)
Answer.
\(2\)
(e)
\(\log_5 \left( 32 \right)\)
Answer.
\(2.153\)
(f)
\(\log_5 \left( \sqrt{5}\right)\)
Answer.
\(\frac{1}{2}\)
(g)
\(\log \left( -10 \right)\)
Answer.
Does not exist
Remark 5.3.12.
Notice that in Activity 5.3.11 part (g) you were unable to evaluate the logarithm. Given that exponentials and logarithms are inverses, their domain and range are related. The range of an expnential function is \((0,\infty)\) which becomes the domain of a logarithmic function. This means that the argument of any logarithmic function must be greater than zero.Activity 5.3.13.
Find the domain of the function \(\log_3(2x-4)\text{.}\)
(a)
Set up an inquality that you must solve to find the domain.
Answer.
\(2x-4 \gt 0\)
(b)
Solve the inequality to find the domain. Write your answer in interval notation.
Answer.
\(x \gt 2\) or \((2, \infty)\)
Subsection 5.3.2 Videos
It would be great to include videos down here, like in the Calculus book!
