# Activity: Analyzing the Relationship between Logarithmic ...

Lesson Title: Analyzing the Relationship between a Logarithmic Function and Its Inverse

Lesson Summary: This activity will give students the opportunity to investigate the inverse of the logarithmic function and explore several uses of the logarithmic function and its inverse in real world applications.

The student will determine numerous ordered pairs that satisfy a given logarithmic function, reverse the x and y coordinates, and use regression to determine the function that represents the inverse of the given logarithmic function. The student will then use this relationship to solve problems.

Key Words: inverse, families of functions (quadratic, logarithmic, exponential), regression analysis, best fit

Background knowledge: The student will know how to find the inverse of a linear or quadratic function algebraically, graphically, and numerically. The students will be able to recognize the graphs of polynomial and exponential functions. The student will have been introduced to algebraic properties of logarithms. The student should have a general knowledge of graphing, table usage, and regression analysis with a graphing calculator.

OAC Standard(s) Addressed: Patterns, Functions, and Algebra

Benchmarks: 8-10 C, D, E; 11-12 A

Grade Level Indicators: 10.10, 11.3, 11.5, 11.6

Learning Objectives: The student should demonstrate numerically and graphically that the inverse of a logarithmic function is an exponential function. The student should solve logarithmic and exponential equations using inverse operations, and recognize real world applications that can be modeled with these functions.

Materials: Activity handout, worksheet, and graphing calculator

Suggested procedures: The lesson will be introduced with a teaser regarding a method of measuring the volume at which the human eardrum will rupture. Students should be grouped in pairs, but each should enter and record data individually.

Assessment(s): Formative assessment will consist of the questions and extensions in the activity and the class discussion following its completion. Summative assessment questions should include identifying graphs of exponential and logarithmic functions and using a logarithmic scale (like decibels or the pH scale) to convert data.

Activity: Analyzing the Relationship between a Logarithmic Function and Its Inverse

This activity will be done after studying algebraic functions, their inverses and basic properties of exponential and logarithmic functions.

PART 1: INVESTIGATING A LOGARITHMIC FUNCTION

Consider the following function: f(x) = log(x)

1. Make a graph of f(x) using your graphing calculator.

2. The table below lists a series of nine output values from f(x). Fill in the table by using the graph and the TABLE to estimate the missing x values.

TABLE 1

|X | |

|[pic] |[pic] |

|Now, let TblStart = 55 and ∆Tbl = 1. You can see that 1.75 is a |Let TblStart = 56 and ∆Tbl = 0.1. Once you can be accurate to the |

|y-value between x = 56 and x = 57. |tenths place, choose the x-value. |

3. Create two lists L1 and L2 in the calculator to store the x and f(x) values. Put the x values in L1 and the f(x) values in L2.

4. Generate a scatter plot of (x, f(x)) using the STAT PLOT feature and the two lists, L1 and L2. Do your x values appear to accurately represent f(x)? If not, modify the values so that they better represent the curve.

5. Generate a new STAT PLOT using L2 and L1 instead of L1 and L2. Hypothesize at least two different families of functions to which this new curve might belong:

family 1: __________________________

family 2: __________________________

6. Using L2 and L1, test your hypothesis with the different regression analysis options and record your results, including the regression equations and R2 values, below.

7. Which of the above selections do you think represents the best fit? Why?

8. If you selected Exponential, you were CORRECT! Congratulate yourself.

9. Look at the exponential equation that you generated. Using your mathematical knowledge and some rounding, estimate g(x), the actual equation of the inverse of f(x):

PART 2: Investigating the Inverse

10. Consider the function h(x) = x3. Solve for x if h(x) = 8.

11. What is i(x), the inverse of h(x)?

12. Will i(x) give all the solutions to h(x) = y for any value of y? Explain your answer.

13. Consider the function h(x) = x2. Solve for x if h(x) = 9.

14. Consider the domain of h(x). What would the range of the inverse be? Is there a function, i(x), the inverse of h(x)?

15. Notice that [pic]will not give all the solutions to h(x)=9. Explain why [pic] is not the inverse of h(x).

Think about the relationship between a function and its inverse. Use that relationship to solve the following:

16. Given: f(x) = log(x). If f(x) = 1.2, find x.

17. Given: g(x) = 10x. If g(x) = 37, find x.

part 3: An Application of Logarithms: Hearing and Decibels

The decibel is the unit used to measure the intensity of a sound. The decibel is named after Alexander Graham Bell who did much work in the area of sound and loudness. Bell discovered that to obtain a sound that seemed twice as loud as another sound, the intensity (how much sound energy per unit area per second hits the eardrum) of the sound must be multiplied by 10. We call this apparent loudness the "intensity level." The following equation is used to compute the intensity of sound:

dB = 10 log (I / Io)

where dB represents decibels, I is the intensity of the sound in question (measured in joules per second per square meter), and Io (read as "I naught" ) is the softest sound the human ear can distinguish, being 10-12 joules per second per square meter. Note that "joules per second per square meter" can be expressed more simply as "Watts per square meter" and is written as "W/m2."

What exactly does "10-12 joules per second per square meter" (or, more simply, 10-12 W/m2) mean? This intensity corresponds to a sound which will displace particles of air by a mere one-billionth of a centimeter. The human ear can detect such a sound! This faintest sound which the human ear can detect is known as the "threshold of hearing" (TOH). The most intense sound which the ear can safely detect without suffering any physical damage is more than one billion times more intense than the threshold of hearing!

It is for this reason that a logarithmic equation is used to measure the intensity of sound. Logarithmic scales are often used when a range of values is extremely large, which is true in the case of the range of intensities that the human ear can detect. Consider this: the human ear can detect a sound as soft as a whisper to as loud as a jet engine flying overhead. That's a big difference in intensity!

(Source: “Sonic Booms and Logarithms,” Robin A. Ward, California Polytechnic State University-San Luis Obispo)

Exercises:

A. Complete the worksheet below.

|Decibels and Logarithms |

| |

|Names:_______________________________________________ |

| |

|Directions: Complete the table using your knowledge of logarithms and the formula: |

| |

|[pic], where I = Intensity |

|Sound Source |Intensity |Intensity Level (dB) |# Times greater than |

| |(W/m2) | |TOH |

|Threshold of hearing (TOH) |[pic] | | |

|Rustling leaves |[pic] |10 | |

|Whisper |[pic] |20 | |

|Normal conversation | |60 | |

|Busy street traffic | |70 | |

|Vacuum cleaner | |80 | |

|Hearing damage possible |[pic] |85 | |

|Lawn mower |[pic] | | |

|Front row at a concert |[pic] |110 | |

|Thunderclap (near) |[pic] | | |

|Threshold of pain |[pic] |130 | |

|Jet take-off |[pic] |140 | |

|Shotgun |[pic] | | |

|Instant perforation of ear drum |[pic] |160 | |

(Source: “Sonic Booms and Logarithms,” Robin A. Ward, California Polytechnic State University-San Luis Obispo)

B. Summarize the concepts that were covered in this activity in your own words. Include a discussion about the inverse functions in Parts 1 and 2, and the reading and exercises involving decibels. Put a star next to any concept that was new to you.

Extension:

1. Given the formula [pic] , where [H+ ]is the concentration (in moles per liter) of hydrogen ions:

Cindy the saboteur wants to make the pool close so that she won’t have to go to swimming practice tomorrow morning. For the pool to remain open, the pH level must be between 7.2 and 7.8. After adding her concoction to the pool, Cindy was able to get a concentration of hydrogen ions of 7.5 × 10-8 moles per liter. Did she succeed in making the pool unsafe enough to close?

2. Given the formula for Richter Scale, [pic], where A is the measured intensity of the earthquake, A0 is the reference intensity, and M is the Richter Scale reading, find the following:

Nancy is a scientist who measured the intensity of an earthquake to be 121,000 times the reference intensity. If Nancy needs to report a Richter scale reading to Daniel, a newspaper reporter, what number should Nancy tell Daniel?

3. Find another application for exponential functions or logarithms that was not mentioned in this lesson. What is the base of the function? Be prepared to share a brief explanation of how they are used to the class.

SOLUTIONS

TABLE 1 – x values should be approximately as follows:

|x |1.12 |2.14 |

|[pic] |[pic] |[pic] |

|(L2,L1) | | |

9. [pic]

10. x = 2

11. [pic] or [pic]

12. Yes, because [pic] is defined for all possible outputs of [pic]. In other words, both functions are defined for all real numbers.

13. [pic]

14. No, if you let [pic], then [pic], and so [pic] is only the inverse of [pic] for [pic].

15. The inputs (x-values) of [pic] can be positive or negative, but the outputs (y-values) of [pic] are only non-negative. The x’s and y’s of these two functions cannot always be “swapped” to get the other function.

16. x = 15.85

17. x = 1.568

|Decibels and Logarithms |

| |

|Names:_______________________________________________ |

| |

|Directions: Complete the table using your knowledge of logarithms and the formula: |

| |

|[pic], where I = Intensity |

|Sound Source |Intensity |Intensity Level (dB) |# Times greater than |

| |(W/m2) | |TOH |

|Threshold of hearing (TOH) |[pic] |0 |0 |

|Rustling leaves |[pic] |10 |10 |

|Whisper |[pic] |20 |100 |

|Normal conversation |10-6 |60 |1,000,000 |

|Busy street traffic |10-7 |70 |10,000,000 |

|Vacuum cleaner |10-4 |80 |100,000,000 |

|Hearing damage possible |[pic] |85 |316,000,000 |

|Lawn mower |[pic] |90 |109 |

|Front row at a concert |[pic] |110 |1011 |

|Thunderclap (near) |[pic] |120 |1012 |

|Threshold of pain |[pic] |130 |1013 |

|Jet take-off |[pic] |140 |1014 |

|Shotgun |[pic] |150 |1015 |

|Instant perforation of ear drum |[pic] |160 |1016 * |

* - Note that 1016 is equal to 10,000,000,000,000,000 or 10 quadrillion.

Extension

1. pH = –log[7.5 x 10-8] = 7.12 Yes, the saboteur succeeded, the pool was unsafe.

2. M = log(121,000) = 5.08

3. Answers will vary. This is desirable so that students do research and find things to add to the discussion that you may not have found or thought of.

Some possibilities include:

• Radioactive decay (base ½ for half-life)

• Other types of exponential growth or decay (perhaps with base e)

• Musical scales (octaves)

• Scientists use logarithms to “linearize” data that are growing exponentially

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