Standard #: MA.912.AR.5.2


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Solve one-variable equations involving logarithms or exponential expressions. Interpret solutions as viable in terms of the context and identify any extraneous solutions.


General Information

Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Algebraic Reasoning
Standard: Write, solve and graph exponential and logarithmic equations and functions in one and two variables.
Date Adopted or Revised: 08/20
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Base of an exponent 
  • Exponent (exponential form) 
  • Exponential function
 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In Algebra I, students worked with the properties of exponents and solved one-variable linear, quadratic and exponential equations. In Math for College Algebra, students will learn that there are similar rules for logarithms and when some exponential equations can’t be solved, one will use logarithms to help convert the bases. In Math for College Algebra, students are building on their skills to solve equations involving exponents or logarithms. In later courses, students will use derivatives to solve different types of function problems. 
  • Instruction includes uses various methods to solve one-variable exponential equations. 
    • For example, students can use the properties of exponents or logarithms to determine the solution for the equation 2x = 32. 
      • Students can rewrite the equation as 2x = 25, noticing that since the base of both sides of equations are the same, then the exponents must be equivalent to one another. Therefore, x = 5. 
      • Students can rewrite the equation as log232 = x, noticing that they can use log 32 the change of base formula to determine the value of x. Therefore, x = log 32log 2, which is equal to 5. 
  • Instruction includes solving logarithmic equations by changing from a logarithmic expression to an exponential expression. 
    • For example, when solving log3(4x − 7) = 2, we can obtain an exact solution by changing the logarithm to exponential form creating the equivalent equation 4x − 7 = 3². Students should recognize that this is a one-variable linear equation 4x − 7 = 9, and therefore, x = 4.
 

Common Misconceptions or Errors

  • Students may not understand the relationship of logarithm and exponential functions and may treat them as separate concepts. 
  • Students may forget to check for extraneous solutions. 
    • For example, when asked to solve logx 36 = 2, students can solve using the square method and get ±6. The base of the logarithm is always positive, so we discard the −6.
 

Instructional Tasks

Instructional Task 1 (MTR.7.1) 
  • Between 7:00 a.m. and 9:00 a.m. cars arrive at Starbucks drive-thru at the rate of 24 cars per hour (0.40 car per minute). The following function can be used to determine the probability that a car will arrive within t minutes of 7:00 a.m. 
    C(t) = 1 − e−0.2t 
    • Part A. Determine the probability that a car will arrive within 5 minutes of 7 a.m. (that is, before 7:05 a.m.). 
    • Part B. Determine the probability that a car will arrive within 30 minutes of 7 a.m. 
    • Part C. What does the value C(t) approach as t becomes unbounded in the positive direction? 
    • Part D. Graph C(t) using graphing technology.
 

Instructional Items

Instructional Item 1 
  • Determine the value of x that satisfies the equation 34x-1 = 127 

Instructional Item 2 
  • Solve for x in the equation 2x = 11. 

Instructional Item 3 
  • Solve for x in the equation log6(2x − 3) = log6(x + 2). 

Instructional Item 4 
  • Determine the value of x that satisfies the equation 2log10 6 − 13 log10 27 = log10x. 

Instructional Item 5 
  • Solve for x in the equation log2(x) = 3.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.




Related Courses

Course Number1111 Course Title222
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2024, 2024 and beyond (current))
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 - 2024, 2024 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200710: Mathematics for College Algebra (Specifically in versions: 2022 - 2024, 2024 and beyond (current))


Related Access Points

Access Point Number Access Point Title
MA.912.AR.5.AP.2 Solve one-variable equations involving logarithms or exponential expressions. Identify any extraneous solutions.


Related Resources

Lesson Plan

Name Description
You’re Pulling My Leg – or Candy!

Students will explore how exponential growth and decay equations can model real-world problems. Students will also discover how manipulating the variables in an exponential equation changes the graph. Students will watch a Perspectives Video to see how exponential growth is modeled in the real world.

Problem-Solving Tasks

Name Description
Algae Blooms

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

Snail Invasion

The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part.

Rising Gas Prices - Compounding and Inflation

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Newton's Law of Cooling

The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.

Carbon 14 Dating in Practice II

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

Bacteria Populations

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.

Student Resources

Problem-Solving Tasks

Name Description
Algae Blooms:

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

Snail Invasion:

The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Newton's Law of Cooling:

The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.

Carbon 14 Dating in Practice II:

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

Bacteria Populations:

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.



Parent Resources

Problem-Solving Tasks

Name Description
Algae Blooms:

In this example, students are asked to write a function describing the population growth of algae. It is implied that this is exponential growth.

Snail Invasion:

The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part.

Rising Gas Prices - Compounding and Inflation:

The purpose of this task is to give students an opportunity to explore various aspects of exponential models (e.g., distinguishing between constant absolute growth and constant relative growth, solving equations using logarithms, applying compound interest formulas) in the context of a real world problem with ties to developing financial literacy skills. In particular, students are introduced to the idea of inflation of prices of a single commodity, and are given a very brief introduction to the notion of the Consumer Price Index for measuring inflation of a body of goods.

Newton's Law of Cooling:

The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.

Carbon 14 Dating in Practice II:

This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.

Bacteria Populations:

This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.



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