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Precalculus MATH F156X Summer 2026

Dr. Latrice Bowman

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Section EL-2 Exponential and Logarithmic Equations

Worksheet Exponential and Logarithmic Equations

For this outcome Specifically, for this outcome, students should be able to:
  • Given an exponential equation where the terms have the same base, solve for the variable using exponential properties.
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  • Given an exponential equation where the terms do not have the same base, solve for the variable using logarithms.
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  • Given a logarithmic equation in which all terms are logarithms, solve for the variable using properties of logarithms.
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  • Given a logarithmic equation in which terms are not all logarithms, solve for the variable using the exponential-logarithmic equivalence.
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  • Use algebraic techniques to solve other types of exponential and logarithmic equations.
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  • Check solutions to ensure they are valid within the domain of the original equation.
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  • Recognize and solve equations that are quadratic in form.
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Exponential Equations where the terms have the same base
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When solving exponential equations where all terms have the same base, or can be rewritten to have the same base, we can set the exponents equal to each other. For example, if we have \(2^x = 2^3\text{,}\) then \(x = 3\text{.}\)
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we can set the exponents equal to each other. For example, if we have \(a^x = a^y\text{,}\) then \(x = y\text{.}\)
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Exponential Equations where the terms have different bases
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When solving exponential equations where all terms do not have the same base, we can take the logarithm of both sides of the equation. For example, if we have \(2^x = 3\text{,}\) we can take the natural logarithm of both sides to get \(x \ln(2) = \ln(3)\text{,}\) and then solve for \(x\text{.}\)
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Logarithmic Equations with all terms containing logarithms
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When solving logarithmic equations where all terms contain logarithms, we can use properties of logarithms to combine the logarithmic terms into a single logarithm, and then solve for the variable. For example, if we have \(\log_a(x) + \log_a(y) = \log_a(z)\text{,}\) we can combine the left-hand side to get \(\log_a(xy) = \log_a(z)\text{,}\) and then solve for \(x\) by setting \(xy = z\text{.}\)
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Logarithmic Equations where terms are not all logarithms
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When solving logarithmic equations where terms are not all logarithms, we can use the exponential-logarithmic equivalence. For example, if we have \(\log_a(x) = y\text{,}\) we can rewrite this as \(x = a^y\text{.}\)
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Worksheet EL-2 Exponential and Logarithmic Functions

Exponential Equations with Same Bases
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Exponential Equations with Different Bases
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Logarithmic Properties
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Logarithmic Equations all Log Terms
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Logarithmic Equations at Least One Non-Log Term
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Other Exponential and Logarithmic Equations
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Section EL-2 Rubric

Worksheet Worksheet

Before attempting the EL-2 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 20. EL2 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.
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.
Solve exponential equations. Correctly identified Type I versus Type II equations and solved accordingly. Either equated the bases and exponents to solve or used natural logarithms to rewrite the equation and solve for x. Either equated the bases and exponents to solve or used natural logarithms to rewrite the equation and solve for x. Minor algebraic errors or did not use the correct method for the type of exponential equation. Multiple errors in solving for the variable (incorrect base, algebraic errors, incorrect method for given type). No work given towards a solution to the exponential equation.
Solve logarithmic equations. Correctly identified Type I versus Type II equations and solved accordingly. Either equated the logarithms or used logarithmic properties to solve for x. Used correct base and domain. Either equated the logarithms or used logarithmic properties to solve for x. Minor algebraic errors or did not use the correct properties. Did not use the correct base or didn’t check the domain. Multiple errors in solving for the variable (incorrect base, algebraic errors, incorrect method for given type). No work given towards a solution to the logarithmic equation.
Solve other exponential and logarithmic equations Used algebraic or logarithmic properties to rewrite and solve the equation. Used correct bases and domains. Minor algebraic or logarithmic property errors when rewriting and solving the equation. Multiple errors in solving for the variable (incorrect base, algebraic errors, incorrect properties for given type). No work given towards a solution to the exponential or logarithmic equation.
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Worksheet EL-2 Exponential and Logarithmic Equations

This outcome covers exponential and logarithmic equations. You should be able to solve exponential and logarithmic equations algebraically and graphically. 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 2 problems. Here are samples of the types of problems you will encounter:
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Solve \(\log_2(2-x)+\log_2(x)=\log_2(5)+\log_2(x-2)\text{.}\)
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Hint.
Use the properties of logarithms to combine the terms on either side of the equation.
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Answer.
There is no solution.
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Solution.
First, we combine the logarithms on each side:
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\(\log_2((2-x)x)=\log_2(5(x-2))\)
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This implies:
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\((2-x)x=5(x-2)\)
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\(2x-x^2=5x-10\)
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\(x^2+3x-10=0\)
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Factoring gives:
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\((x+5)(x-2)=0\)
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So \(x=-5\) or \(x=2\text{.}\) However, both values are outside the domain of the original equation, so there is no solution.
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(c)
Solve \(8^{2x}-2(8^x)-8=0\text{.}\)
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Hint.
Let \(u=8^x\) and rewrite the equation in terms of \(u\text{.}\)
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Answer.
\(x=\frac{2}{3}\)
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Solution.
Let \(u=8^x\text{.}\) Then the equation becomes:
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\(u^2-2u-8=0\)
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Factoring gives:
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\((u-4)(u+2)=0\)
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So \(u=4\) or \(u=-2\text{.}\) Since \(u=8^x\text{,}\) we have:
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\(8^x=4\) or \(8^x=-2\)
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The second equation has no real solution, so we consider the first:
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\(8^x=4\)
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Rewriting both sides in terms of base 2:
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\((2^3)^x=2^2\)
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\(2^{3x}=2^2\)
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\(3x=2\)
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\(x=\frac{2}{3}\)
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