MA.912.NSO.1.7

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

• MA.912.AR.5.2
• MA.912.AR.5.8
• MA.912.AR.5.9

Terms from the K-12 Glossary

• Inverse function

Vertical Alignment

Previous Benchmarks

• MA.8.NSO.1.3

Next Benchmarks

• MA.912.C.2.8

Purpose and Instructional Strategies

• 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 log_1.03 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 Tasks

• 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? 

• Use the functions below to answer the following questions.  
  
  $f$($x$) = 10^0.2$x$
  $h$($x$) = 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.

• Use the properties of logarithms to rewrite each expression into lowest terms. 
  • Part A. log₄ 4$x$²
    
    
  • Part B. ln $\frac{\text{<math><mi>x</mi><mi>y</mi></math>}}{\text{<math><mi>z</mi></math>}}$

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

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