# MA.912.NSO.1.7

Given an algebraic logarithmic expression, generate an equivalent algebraic expression using the properties of logarithms or exponents.

### Clarifications

Clarification 1: Within the Mathematics for Data and Financial Literacy Honors course, problem types focus on money and business.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Number Sense and Operations
Date Adopted or Revised: 08/20
Status: State Board Approved

## Benchmark Instructional Guide

### Terms from the K-12 Glossary

• Inverse function

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 8, students generated equivalent numerical expressions and evaluated expressions using the Laws of Exponents with integer exponents. In Algebra I, students worked with rational-number exponents. In Mathematics for College Algebra, students extend the Laws of Exponents to the Properties of Logarithms.
• Instruction makes the connection between the properties of logarithms and the properties of exponents. Explain that a logarithm is defined as an exponent establishing the equivalence of $y$$a$$x$ and, $x$ = log$a$$y$ given $a$ > 0 and $a$ ≠ 1 (MTR.5.1). Remind students that logarithmic and exponential operations are inverse operations, so the properties of the logarithms are the “opposite” of the properties of the exponents.

• Instruction includes making the connection between the change of base formula and the inverse relationship between exponents and logarithms.
• For example, students should know that by definition $b$log$b$ $x$ = $x$. Therefore, students can take the log with base $a$ of both sides of the equation to obtain log$a$ $b$log$b$ $x$ = log$a$ $x$. Then, students can use the the Power Property to rewrite the equation as (log$b$ $x$)(log$a$ $b$) = log$a$ $x$. Students should notice that there the arguments for two of the logs are $b$ and $x$ with each the same base of $a$. So, one can divide both sides of the equation by log$a$ $b$  to isolate log$b$ $x$ obtaining
• Instruction encourages students to read the logarithmic expressions and then discuss the meaning before trying to evaluate them or use the properties (MTR.4.1)
• For example, present students with log1.03 1.092727 and ask them for its meaning, “the exponent required on the base 1.03 to obtain 1.092727.”
• Students should have practice combining the properties of logarithms to generate equivalent algebraic expressions by either simplifying (condensing) or expanding the expression.
• Problem types include logarithms with different bases, including common logarithms and natural logarithms.

### Common Misconceptions or Errors

• Students tend to treat “log” as a variable rather than as an operation:
• For example, they may see “log” as a common factor in the expression log $x$ + log $y$ and mistakenly write log($x$ + $y$).
• For example, they may distribute the “log” in the expression log($a$ · $b$) and rewrite it as log $a$ · log $b$ instead of log$a$ $m$ + log$a$ $n$. Similarly, they may incorrectly rewrite log ($\frac{\text{a}}{\text{b}}$) as $\frac{\text{log a}}{\text{log b}}$
• For example, they may divide both sides of the equation log(7$x$ − 12) = 2log $x$ by “log” to mistakenly obtain 7$x$ − 12 = 2$x$
• Students may cancel the “log” from the numerator and the denominator in an expression. Remind students that just like with roots we can simplify or expand logarithms if the argument is fully factored.

Instructional Task 1 (MTR.7.1
• The Loudness of Sound formula, measure in decibels ($d$$B$), is $L$ = 10 log $I$, where $L$ is the loudness, and $I$ is the intensity of sound.
• Part A. The formula Δ$L$ = $L$2 - $L$1, describe the sound intensity level between two loudness, $L$1 and $L$2. Write the formula in terms of the intensity of the sounds, $I$2 and $I$1
• Part B. Rewrite the formula for the sound intensity level, Δ$L$, as a single logarithm.
• Part C. The table below shows the Ratios of Intensities and Corresponding Differences in Sound Intensity Levels. Using the Properties of Logarithms show that if a sound is 100 times as intense as another, it has a sound level about 20 dB higher.

• Part D. If the lowest or threshold intensity of sound a person with normal hearing can perceive is $x$0 = 10−12 $W$ ⁄ $m$2, what is the intensity of sound at a pain level of 120 dB?

Instructional Task 2 (MTR.4.1, MTR.5.1
• Use the functions below to answer the following questions.
$f$($x$) = 100.2$x$
$h$($x$) = 5(log $x$)
• Part A. Use technology to graph the functions  $f$ and $h$ on the same coordinate plane. What do you notice?
• Part B. Determine $f$(2) and $h$(2). What do you notice?
• Part C. Discuss with a partner why logarithmic functions are said to be the inverse of exponential functions.

### Instructional Items

Instructional Item 1
• Use the properties of logarithms to rewrite each expression into lowest terms.
• Part A. log4 4$x$2

• Part B. ln $\frac{\text{xy}}{\text{z}}$

Instructional Item 2
• Write each expression as a single logarithmic quantity:
• Part A. 3ln $x$ + 4ln $y$ − 5ln
• Part B. $\frac{\text{3}}{\text{2}}$log2 $x$6$\frac{\text{3}}{\text{4}}$log2 $x$8

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

## Related Courses

This benchmark is part of these courses.
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))
1200388: Mathematics for Data and Financial Literacy Honors (Specifically in versions: 2022 and beyond (current))
1200710: Mathematics for College Algebra (Specifically in versions: 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.NSO.1.AP.7: Given an algebraic logarithmic expression, identify an equivalent algebraic expression using the properties of logarithms or exponents.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

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.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

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.