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Precalculus MATH F156X Summer 2026

Dr. Latrice Bowman

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Section EL-1 Exponential and Logarithmic Functions

Worksheet Exponential and Logarithmic Functions

For this outcome Specifically, for this outcome, students should be able to:
  • Given an exponential function, identify its domain, range, and asymptotes.
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  • Given a logarithmic function, identify its domain, range, and asymptotes.
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  • Graph exponential and logarithmic functions.
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  • Use properties of logarithms to simplify expressions.
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  • Given the graph of an exponential function, write the equation of the function.
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  • Given the graph of a logarithmic function, write the equation of the function.
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Exponential Functions
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An exponential function is a function of the form \(f(x) = a^x\text{,}\) where \(a \gt 0\) and \(a \neq 1\text{.}\) The base \(a\) is a constant, and the exponent is the variable \(x\text{.}\) Exponential functions are used to model growth and decay.
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Logarithmic Functions
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A logarithmic function is the inverse of an exponential function. It is written as \(f(x) = \log_a(x)\text{,}\) where \(a \gt 0\text{,}\) \(a \neq 1\text{,}\) and \(x \gt 0\text{.}\) Logarithmic functions are used to solve exponential equations and model phenomena that grow or decay exponentially.
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Properties of Exponential and Logarithmic Functions
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Logarithmic functions are the inverse of exponential functions. The following properties are important to remember:
  • \(\displaystyle \log_a(a^x) = x\)
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  • \(\displaystyle a^{\log_a(x)} = x\)
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  • \(\displaystyle \log_a(xy) = \log_a(x) + \log_a(y)\)
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  • \(\displaystyle \log_a(\frac{x}{y}) = \log_a(x) - \log_a(y)\)
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  • \(\displaystyle \log_a(x^n) = n \cdot \log_a(x)\)
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  • \(\displaystyle \log_a(1) = 0\)
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  • \(\displaystyle \log_a(a) = 1\)
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  • \(\log x\) can be written as \(\log_{10} x\) and is called the common logarithm.
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  • \(\ln x\) can be written as \(\log_e x\) and is called the natural logarithm.
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  • The change of base formula allows you to compute logarithms with any base using a calculator: \(\log_a(x) = \frac{\log_b(x)}{\log_b(a)}\) for any positive \(b \neq 1\text{.}\)
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Graphs of Exponential and Logarithmic Functions
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To graph an exponential function:
  • The domain is all real numbers.
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  • The range is all real numbers greater than 0.
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  • The x-axis is a horizontal asymptote.
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  • If \(a \gt 1\text{,}\) the graph rises from left to right and approaches the x-axis as \(x\) approaches negative infinity.
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  • If \(0 \lt a \lt 1\text{,}\) the graph falls from left to right and approaches the x-axis as \(x\) approaches positive infinity.
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  • The y-intercept is at (0, 1) and there are no x-intercepts.
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  • The graph is continuous and one-to-one.
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To graph a logarithmic function:
  • The domain is all real numbers greater than 0.
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  • The range is all real numbers.
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  • The y-axis is a vertical asymptote.
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  • If \(a \gt 1\text{,}\) the graph rises from left to right and approaches the y-axis as \(x\) approaches 0 from the right.
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Worksheet EL-1 Exponential and Logarithmic Functions

Exponential Functions
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Graphs of Exponential Functions
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Logarithmic Functions
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Graphs of Logarithmic Functions
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Section EL-1 Rubric

Worksheet Worksheet

Before attempting the EL-1 outcome, you should review the following rubric. This shows the criteria you will be graded on and the expectations to earn each mastery level.
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Table 19. EL1 Rubric
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Criteria M P R N
Simplifying Algebraic Expressions All answers are fully simplified. Exact answers are given, not approximations. Radical and exponential expressions are used appropriately and are simplified in the correct manner No algebraic errors OR some expressions are not simplified. There are some algebraic errors and some expressions were not simplified. There are some algebraic errors and some expressions were not simplified.
Basics of Graphing Graph is labeled with units for each axis. The graph includes all of the important features (intercepts, asymptotes, deleted points, etc).  The graph is appropriately drawn to accommodate all features. Graph is labeled with units for each axis and includes most of the important features or the graph has all of the important features but is not labeled. Graph is not labeled and has some of the important features. No graph was given or the given graph is not labeled and does not have the correct features of the function.
Problem Structure and Mathematical Communication Problems have a clear beginning, middle, and end. Work progresses clearly from one step to the next. The work provided uses appropriate methods and notation and provides a clear solution. Problems have clear beginnings and ends. Work progresses from one step to the next. The work provided uses algebraic methods, correct notation, and provides a solution. Parts of the mathematical structure are missing, or steps in the progress of the solution are missing. There are errors in the mathematical notation. The work provided uses algebraic methods and provides a solution. There is not a clear structure to the solution, notation is used incorrectly, and it is unclear what methods are being used.
Find exponential and logarithmic attributes. Provided the correct, simplified domain/range using correct interval notation; both x and y intercepts were given in correct notation; Used appropriate methods to find and label the vertical or horizontal asymptote with correct notation. Provided the a domain/range using correct interval notation; both x and y intercepts; Used appropriate methods to find and label the vertical or horizontal asymptote. Provided a domain and range; provided points. Not enough evidence to show knowledge of domain, range, intercepts, and asymptotes.
Graph an exponential and logarithmic function. Graph contains all relevant information; has correct end-behavior, asymptotes and intercepts. Graph contains most of the relevant information; has partially correct end-behavior, asymptotes and intercepts. A graph was given but it does not represent an exponential or logarithmic function. No graph was given.
Find formula for exponential/logarithmic function. Based on graph, set up the correct function type; found the vertical shift, horizontal shift, any reflections; found the correct base; gave the correct equation. Use the asymptotes and given points to find the function. Based on graph, set up the correct function type; Use the asymptotes and given points to find a function. Used the given information to find an exponential/logarithmic function. No exponential/logarithmic function given.
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Worksheet EL-1 Exponential and Logarithmic Functions

This outcome covers exponential and logarithmic functions and models. You should be able to rewrite exponential functions in various forms. Be able to find the domain, range, and intercepts of the function. You should also be able to graph exponential and logarithmic functions. Here you will also be asked to setup and solve real-world problems using exponential and logarithmic models. You should be able to write out complete steps to solve these types of problems by hand, without the use of a calculator.
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For the quiz on this outcome you would be given 30 minutes to complete 3 problems. Here are samples of the types of problems you will encounter:
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1.

Given \(f(x)=\left(\frac{3}{4}\right)^{x+2}-3\text{.}\)
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(a)
State the domain and range.
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Hint.
Recall that the domain of an exponential function is all real numbers, and the range depends on the vertical shift.
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Answer.
Domain: \((-\infty, \infty)\text{;}\) Range: \((-3, \infty)\)
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Solution.
The domain of an exponential function is always all real numbers. The range depends on the vertical shift. Since the function is shifted down by 3 units, the range is \((-3, \infty)\text{.}\)
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(b)
State the intercepts.
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Hint.
To find the y-intercept, set \(x=0\text{.}\) To find the x-intercept, set \(y=0\text{.}\)
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Answer.
y-intercept: \((0, -\frac{39}{16})\text{;}\) x-intercept: \((\frac{\ln3}{\ln\frac{3}{4}}+2,0)\)
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Solution.
To find the y-intercept, set \(x=0\text{:}\) \(f(0)=\left(\frac{3}{4}\right)^{0+2}-3 =\left(\frac{3}{4}\right)^{2}-3=\frac{9}{16}-3=-\frac{39}{16}\text{.}\) So the y-intercept is \((0, -\frac{39}{16})\text{.}\) To find the x-intercept, set \(y=0\text{:}\) \(0=\left(\frac{3}{4}\right)^{x+2}-3\text{.}\) \(\left(\frac{3}{4}\right)^{x+2}=3\text{.}\) Taking the natural logarithm of both sides and dividing gives \(x+2=\frac{\ln3}{\ln\frac{3}{4}}\text{.}\) Therefore, the x-intercept is \(\left(\frac{\ln3}{\ln\frac{3}{4}}+2,0\right)\text{.}\)
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(c)
State the asymptote(s).
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Hint.
Exponential functions have horizontal asymptotes, while logarithmic functions have vertical asymptotes.
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Answer.
Horizontal asymptote: \(y=-3\)
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Solution.
Exponential functions have horizontal asymptotes determined by the vertical shift. Since the function is shifted down by 3 units, the horizontal asymptote is \(y=-3\text{.}\)
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(d)
Graph the function.
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Hint.
Plot the intercepts and use the asymptote to guide the shape of the graph.
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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
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2.

Given \(y=4-\log_2(x+3)\text{.}\)
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(a)
Rewrite the function in exponential form.
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Hint.
Use the exponential-logarithmic equivalence to rewrite the function.
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Answer.
\(x=2^{4-y}-3\)
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Solution.
To rewrite the function in exponential form, we start with the given logarithmic function and begin by isolating the logarithmic term \(4-y=\log_2(x+3)\text{.}\) Then, we apply the exponential-logarithmic equivalence and solve for \(x\text{:}\)
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\(x+3=2^{4-y}\)
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\(x=2^{4-y}-3\)
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(b)
State the domain and range.
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Hint.
Consider the restrictions on the input and output values.
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Answer.
Domain: \((-3, \infty)\text{;}\) Range: \((-\infty, \infty)\)
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Solution.
The domain of the logarithmic function is determined by the requirement that the argument of the logarithm be positive. Thus, \(x+3>0\text{,}\) which implies \(x>-3\text{.}\) The range of the logarithmic function is all real numbers, so \((-\infty, \infty)\text{.}\)
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(c)
State the intercepts.
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Hint.
To find the y-intercept, set \(x=0\) and solve for \(y\text{.}\) To find the x-intercept, set \(y=0\) and solve for \(x\text{.}\)
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Answer.
y-intercept: \((0, 4-\frac{\ln 3}{\ln 2})\text{;}\) x-intercept: \((13, 0)\)
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Solution.
To find the y-intercept, set \(x=0\) and solve for \(y\) and give answer in terms of natural logarithms:
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\(y=4-\frac{\ln 3}{\ln 2}\)
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To find the x-intercept, set \(y=0\) and solve for \(x\text{:}\)
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\(0=4-\log_2(x+3)\)
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\(\log_2(x+3)=4\)
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\(x+3=2^4\)
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\(x=2^4-3\)
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\(x=13\)
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(d)
State the asymptote(s).
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Hint.
To find the vertical asymptote of the logarithmic function, consider the values of \(x\) for which the argument of the logarithm is zero.
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Answer.
Vertical asymptote: \(x=-3\)
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Solution.
The vertical asymptote occurs where the argument of the logarithm is zero. Thus, \(x+3=0\text{,}\) which implies \(x=-3\text{.}\)
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(e)
Graph the function.
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Hint.
Plot the intercepts and use the asymptote to guide the shape of the graph.
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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
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3.

(a)
An exponential function has a horizontal asymptote at \(y=-3\text{.}\) The graph of the function passes through the points \((1,1)\) and \((2,5)\text{.}\) If the function has the form \(g(x)=a^{x+b}+ c\) for real numbers \(a, b\text{,}\) and \(c\text{,}\) find \(g(x)\text{.}\)
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Hint.
Use the horizontal asymptote to find \(c\text{,}\) then use the two given points to find \(a\) and \(b\text{.}\)
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Answer.
g(x)=2^{x+1}-3
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Solution.
The horizontal asymptote at \(y=-3\) indicates that \(c=-3\text{.}\) Using the points \((1,1)\) and \((2,5)\text{,}\) we can solve for \(a\) and \(b\text{.}\)
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\(1=a^{1+b}-3\) and \(5=a^{2+b}-3\)
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Adding 3 to both sides of each equation gives: \(4=a^{1+b}\) and \(8=a^{2+b}\)
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Dividing the second equation by the first gives:
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\(\frac{8}{4}=a^{(2+b)-(1+b)}=a\)
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\(a=2\)
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Substituting back into the first equation gives:
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\(4=2^{1+b}\)
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\(2^2=2^{1+b}\)
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\(2=1+b\)
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\(b=1\)
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Thus, the function is \(g(x)=2^{x+1}-3\text{.}\)
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