Skip to main content
Contents Dark Mode Prev Up Next \(\newcommand{\markedPivot}[1]{\boxed{#1}}
\newcommand{\IR}{\mathbb{R}}
\newcommand{\IC}{\mathbb{C}}
\renewcommand{\P}{\mathcal{P}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\RREF}{\operatorname{RREF}}
\newcommand{\vspan}{\operatorname{span}}
\newcommand{\setList}[1]{\left\{#1\right\}}
\newcommand{\setBuilder}[2]{\left\{#1\,\middle|\,#2\right\}}
\newcommand{\unknown}{\,{\color{gray}?}\,}
\newcommand{\drawtruss}[2][1]{
\begin{tikzpicture}[scale=#1, every node/.style={scale=#1}]
\draw (0,0) node[left,magenta]{C} --
(1,1.71) node[left,magenta]{A} --
(2,0) node[above,magenta]{D} -- cycle;
\draw (2,0) --
(3,1.71) node[right,magenta]{B} --
(1,1.71) -- cycle;
\draw (3,1.71) -- (4,0) node[right,magenta]{E} -- (2,0) -- cycle;
\draw[blue] (0,0) -- (0.25,-0.425) -- (-0.25,-0.425) -- cycle;
\draw[blue] (4,0) -- (4.25,-0.425) -- (3.75,-0.425) -- cycle;
\draw[thick,red,->] (2,0) -- (2,-0.75);
#2
\end{tikzpicture}
}
\newcommand{\trussNormalForces}{
\draw [thick, blue,->] (0,0) -- (0.5,0.5);
\draw [thick, blue,->] (4,0) -- (3.5,0.5);
}
\newcommand{\trussCompletion}{
\trussNormalForces
\draw [thick, magenta,<->] (0.4,0.684) -- (0.6,1.026);
\draw [thick, magenta,<->] (3.4,1.026) -- (3.6,0.684);
\draw [thick, magenta,<->] (1.8,1.71) -- (2.2,1.71);
\draw [thick, magenta,->] (1.6,0.684) -- (1.5,0.855);
\draw [thick, magenta,<-] (1.5,0.855) -- (1.4,1.026);
\draw [thick, magenta,->] (2.4,0.684) -- (2.5,0.855);
\draw [thick, magenta,<-] (2.5,0.855) -- (2.6,1.026);
}
\newcommand{\trussCForces}{
\draw [thick, blue,->] (0,0) -- (0.5,0.5);
\draw [thick, magenta,->] (0,0) -- (0.4,0.684);
\draw [thick, magenta,->] (0,0) -- (0.5,0);
}
\newcommand{\trussStrutVariables}{
\node[above] at (2,1.71) {\(x_1\)};
\node[left] at (0.5,0.866) {\(x_2\)};
\node[left] at (1.5,0.866) {\(x_3\)};
\node[right] at (2.5,0.866) {\(x_4\)};
\node[right] at (3.5,0.866) {\(x_5\)};
\node[below] at (1,0) {\(x_6\)};
\node[below] at (3,0) {\(x_7\)};
}
\newcommand{\N}{\mathbb N}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\DeclareMathOperator{\arcsec}{arcsec}
\DeclareMathOperator{\arccot}{arccot}
\DeclareMathOperator{\arccsc}{arccsc}
\newcommand{\tuple}[1]{\left\langle#1\right\rangle}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
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 \(y\text{.}\)
(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.
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
Answer .
\(\dfrac{1}{4}\) \(-2\)
\(\dfrac{1}{2}\) \(-1\)
\(1\) \(0\)
\(2\) \(1\)
\(4\) \(2\)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
(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
\(x \to \infty\text{.}\)
(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
\(x \to -\infty\text{.}\)
(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.
(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.
(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 .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
(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.
(i)
What the equation for each asymptote of \(f(x)\text{?}\) Select all that apply.
(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.
Activity 5.4.5 .
Let
\(f(x)=\log_4 x\text{.}\)
(a) Answer .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
(b)
Match the following functions to their graphs.
\(\displaystyle g(x)= -\log_4 x \)
\(\displaystyle h(x)= \log_4 (-x) \)
\(\displaystyle j(x)= \log_4 (x+1) \)
\(\displaystyle k(x)= \log_4 (x) + 1 \)
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
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 .
\(f(x)\text{:}\)
Range:
\((-\infty,\infty)\)
\(g(x)\text{:}\)
Range:
\((-\infty,\infty)\)
\(h(x)\text{:}\)
Range:
\((-\infty,\infty)\)
\(j(x)\text{:}\)
Range:
\((-\infty,\infty)\)
\(k(x)\text{:}\)
Range:
\((-\infty,\infty)\)
(d)
Which of the transformations affected the domain of the logarithmic function? Select all that apply.
A reflection over the
\(x\) -axis.
A reflection over the
\(y\) -axis.
(e)
Which of the transformations affected the range of the logarithmic function? Select all that apply.
A reflection over the
\(x\) -axis.
A reflection over the
\(y\) -axis.
(f)
Which of the transformations affected the asymptote of the logarithmic function? Select all that apply.
A reflection over the
\(x\) -axis.
A reflection over the
\(y\) -axis.
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 .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Range:
\((-\infty,\infty)\)
(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 .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Transformation: shift right 3
Range:
\((-\infty,\infty)\)
(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 .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Transformation: vertical stretch with factor of three.
Range:
\((-\infty,\infty)\)
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.
(a) Answer .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Range:
\((-\infty,\infty)\)
(b) Answer .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Range:
\((-\infty,\infty)\)
(c)
\(f(x)=\log_{\frac{1}{5}} x\)
Answer .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Range:
\((-\infty,\infty)\)
(d)
\(f(x)=\log_{\frac{1}{3}} x +2\)
Answer .
Diagram Exploration Keyboard Controls
Key
Action
Enter, A
Activate keyboard driven exploration
B
Activate menu driven exploration
Escape
Leave exploration mode
Cursor down
Explore next lower level
Cursor up
Explore next upper level
Cursor right
Explore next element on level
Cursor left
Explore previous element on level
X
Toggle expert mode
W
Extra details if available
Space
Repeat speech
M
Activate step magnification
Comma
Activate direct magnification
N
Deactivate magnification
Z
Toggle subtitles
C
Cycle contrast settings
T
Monochrome colours
L
Toggle language (if available)
K
Kill current sound
Y
Stop sound output
O
Start and stop sonification
P
Repeat sonification output
Range:
\((-\infty,\infty)\)
Subsection 5.4.2 Exercises
