Graphs of Logarithmic Functions (EL4)

Section 5.4 Graphs of Logarithmic Functions (EL4)

Objectives

Graph logarithmic functions and determine the domain, range, and asymptotes.

Subsection 5.4.1 Activities

Activity 5.4.1.

Consider the function \(g(x)=\log_2 x\text{.}\)

(a)

Since we are familiar with graphing exponential functions, we’ll use that to help us graph logarithmic ones. Rewrite \(g(x)\) in exponential form, replacing \(g(x)\) with \(y\text{.}\)

\(\displaystyle x^y=2\)
\(\displaystyle 2^y=x\)
\(\displaystyle y^2=x\)
\(\displaystyle 2^x=y\)
\(\displaystyle x^2=y\)

Answer.

(b)

Fill in the table of values. Notice you are given \(y\)-values, not \(x\)-values to plug in since those are easier in the equivalent exponential form. Then plot the points on a graph.

\(x\)	\(y\)
	\(-2\)
	\(-1\)
	\(0\)
	\(1\)
	\(2\)

Answer.

\(x\)	\(y\)
\(\frac{1}{4}\)	\(-2\)
\(\frac{1}{2}\)	\(-1\)
\(1\)	\(0\)
\(2\)	\(1\)
\(4\)	\(2\)

(c)

What seems to be happening with the graph as \(x\) goes toward infinity? Plug in large positive values of \(x\) to test your guess, then describe the end behavior.

As \(x \to \infty\text{,}\) \(y \to -\infty\text{.}\)
As \(x \to \infty\text{,}\) \(y \to 0\text{.}\)
As \(x \to \infty\text{,}\) \(y \to 6\text{.}\)
As \(x \to \infty\text{,}\) \(y \to \infty\text{.}\)
The graph isn’t defined as \(x \to \infty\text{.}\)

Answer.

(d)

What seems to be happening with the graph as \(x\) goes toward negative infinity? Plug in large negative values of \(x\) to test your guess, then describe the end behavior.

As \(x \to -\infty\text{,}\) \(y \to -\infty\text{.}\)
As \(x \to -\infty\text{,}\) \(y \to 0\text{.}\)
As \(x \to -\infty\text{,}\) \(y \to 6\text{.}\)
As \(x \to -\infty\text{,}\) \(y \to \infty\text{.}\)
The graph isn’t defined as \(x \to -\infty\text{.}\)

Answer.

(e)

What seems to be happening with the graph as we approach \(x\)-values closer and closer to zero from the positive direction?

As \(x \to 0\) from the positive direction, \(y \to -\infty\text{.}\)
As \(x \to 0\) from the positive direction, \(y \to 0\text{.}\)
As \(x \to 0\) from the positive direction, \(y \to \infty\text{.}\)
As \(x \to 0\) from the positive direction, the graph isn’t defined.

Answer.

(f)

What seems to be happening with the graph as we approach \(x\)-values closer and closer to zero from the negative direction?

As \(x \to 0\) from the negative direction, \(y \to -\infty\text{.}\)
As \(x \to 0\) from the negative direction, \(y \to 0\text{.}\)
As \(x \to 0\) from the negative direction, \(y \to \infty\text{.}\)
As \(x \to 0\) from the negative direction, the graph isn’t defined.

Answer.

(g)

Complete the graph you started in Task 5.4.1.a, connecting the points and including the end behavior and behavior near zero that you’ve just determined.

Answer.

(h)

Does your graph seem to have any asymptotes?

No. There are no asymptotes.
There is a vertical asymptote but no horizontal one.
There is a horizontal asymptote but no vertical one.
The graph has both a horizontal and vertical asymptote.

Answer.

(i)

What the equation for each asymptote of \(f(x)\text{?}\) Select all that apply.

There are no asymptotes.
\(\displaystyle x=0\)
\(\displaystyle x=6\)
\(\displaystyle y=0\)
\(\displaystyle y=6\)

Answer.

(j)

Find the domain and range of \(g(x)\text{.}\) Write your answers using interval notation.

Answer.

Domain: \((0,\infty)\text{,}\) Range: \((-\infty, \infty)\)

(k)

Find the interval(s) where \(g(x)\) is increasing and the interval(s) where \(g(x)\) is decreasing. Write your answers using interval notation.

Answer.

Increasing: \((0,\infty)\text{,}\) Decreasing: nowhere

Activity 5.4.2.

The function we’ve just graphed, \(g(x)=\log_2 x\text{,}\) and the function \(f(x)=2^x\) (which we graphed in Activity 5.2.1) are inverse functions.

(a)

How are the graphs of \(f(x)\) and \(g(x)\) similar?

Answer.

Answers could include basic shape (though mirrored), both are increasing.

(b)

How are the graphs of \(f(x)\) and \(g(x)\) different?

Answer.

Answers could include flipped \(x\) and \(y\) values, flipped asymptote, one has \(x\)-intercept and one has \(y\)-intercept, domain and range are swapped.

Remark 5.4.3.

The graph of a logarithmic function \(g(x)=\log_b x\) where \(b>0\) and \(b \neq 1\) has the following characteristics:

Its domain is \((0,\infty)\) and its range is \((-\infty,\infty)\text{.}\)
There is a vertical asymptote at \(x=0\text{.}\) There is no horizontal asymptote.
There is an \(x\)-intercept at \((1,0)\text{.}\) There is no \(y\)-intercept.

Remark 5.4.4.

Just as with other types of functions, we can use transformations to graph logarithmic functions. For a reminder of these transformations, see Section 2.4 and the following definitions:

Activity 5.4.5.

Let \(f(x)=\log_4 x\text{.}\)

(a)

Graph \(f(x)\text{.}\)

Answer.

(b)

Match the following functions to their graphs.

\(\displaystyle g(x)= -\log_4 x \)
\(\displaystyle h(x)= \log_4 (-x) \)
\(\displaystyle j(x)= \log_4 (x+1) \)
\(\displaystyle k(x)= \log_4 (x) + 1 \)

Answer.

\(g(x)= -\log_4 x \) is A.
\(h(x)= \log_4 (-x) \) is C.
\(j(x)= \log_4 (x+1) \) is D.
\(k(x)= \log_4 (x) + 1 \) is B.

(c)

Find the domain, range, and equation of the asymptote for the parent function \(\left(f(x)\right)\) and each of the four transformations \(\left(g(x), h(x), j(x), \text{ and } k(x)\right)\text{.}\)

Answer.

\(f(x)\text{:}\)

Domain: \((0,\infty)\)
Range: \((-\infty,\infty)\)
Asymptote: \(x=0\)

\(g(x)\text{:}\)

Domain: \((0,\infty)\)
Range: \((-\infty,\infty)\)
Asymptote: \(x=0\)

\(h(x)\text{:}\)

Domain: \((-\infty,0)\)
Range: \((-\infty,\infty)\)
Asymptote: \(x=0\)

\(j(x)\text{:}\)

Domain: \((-1,\infty)\)
Range: \((-\infty,\infty)\)
Asymptote: \(x=-1\)

\(k(x)\text{:}\)

Domain: \((0,\infty)\)
Range: \((-\infty,\infty)\)
Asymptote: \(x=0\)

(d)

Which of the transformations affected the domain of the logarithmic function? Select all that apply.

A vertical shift.
A horizontal shift.
A reflection over the \(x\)-axis.
A reflection over the \(y\)-axis.
None of these.

Answer.

B,C

(e)

Which of the transformations affected the range of the logarithmic function? Select all that apply.

A vertical shift.
A horizontal shift.
A reflection over the \(x\)-axis.
A reflection over the \(y\)-axis.
None of these.

Answer.

(f)

Which of the transformations affected the asymptote of the logarithmic function? Select all that apply.

A vertical shift.
A horizontal shift.
A reflection over the \(x\)-axis.
A reflection over the \(y\)-axis.
None of these.

Answer.

Activity 5.4.6.

Consider the function \(f(x)=\ln(x)\text{.}\)

(a)

Graph \(f(x)=\ln(x)\text{.}\) First find \(f(1)\) and \(f(e)\text{.}\) Then use what you know about the characteristics of logarithmic graphs to sketch the rest. Then state the domain, range, and equation of the asymptote. (Recall that \(e \approx 2.72\) to help estimate where to put your points.)

Answer.

Domain: \((0,\infty)\)
Range: \((-\infty,\infty)\)
Asymptote: \(x=0\)

(b)

Sketch the graph of \(g(x)=\ln(x-3)\) using transformations. State the transformation(s) used, the domain, the range, and the equation of the asymptote.

Answer.

Transformation: shift right 3
Domain: \((3,\infty)\)
Range: \((-\infty,\infty)\)
Asymptote: \(x=3\)

(c)

Sketch the graph of \(h(x)=3\ln(x)\) using transformations. State the transformation(s) used, the domain, the range, and the equation of the asymptote.

Answer.

Transformation: vertical stretch with factor of three.
Domain: \((0,\infty)\)
Range: \((-\infty,\infty)\)
Asymptote: \(x=0\)

Activity 5.4.7.

Graph each of the following logarithmic functions. Include any asymptotes with a dotted line. State the domain, the range, and the equation of the asymptote.