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Lesson 10-1 Exponential Functions Lesson 10-2 Logarithms and Logarithmic Functions Lesson 10-3 Properties of Logarithms Lesson 10-4 Common Logarithms Lesson 10-5 Base e and Natural Logarithms Lesson 10-6 Exponential Growth and Decay Contents

Example 1 Graph an Exponential Function Example 2 Identify Exponential Growth and Decay Example 3 Write an Exponential Function Example 4 Simplify Expressions with Irrational Exponents Example 5 Solve Exponential Equations Example 6 Solve Exponential Inequalities Lesson 1 Contents

x Sketch the graph of . Then state the function’s domain and range. Make a table of values. Connect the points to sketch a smooth curve. 16 2 4 1 –1 –2 x Example 1-1a

Answer: The domain is all real numbers, while the range is all positive numbers. Example 1-1b

Sketch the graph of Then state the function’s domain and range. Answer: The domain is all real numbers; the range is all positive numbers. Example 1-1c

Determine whether represents exponential growth or decay. Answer: The function represents exponential decay, since the base, 0.7, is between 0 and 1. Example 1-2a

Determine whether represents exponential growth or decay. Answer: The function represents exponential growth, since the base, 3, is greater than 1. Example 1-2b

Determine whether represents exponential growth or decay. Answer: The function represents exponential growth, since the base, is greater than 1. Example 1-2c

Answer: The function represents exponential decay, since the Determine whether each function represents exponential growth or decay. a. b. c. Answer: The function represents exponential decay, since the base, 0.5, is between 0 and 1. Answer: The function represents exponential growth, since the base, 2, is greater than 1. Answer: The function represents exponential decay, since the base, is between 0 and 1. Example 1-2d

Cellular Phones In December of 1990, there were 5,283,000 cellular telephone subscribers in the United States. By December of 2000, this number had risen to 109,478,000. Write an exponential function of the form that could be used to model the number of cellular telephone subscribers y in the U.S. Write the function in terms of x, the number of years since 1990. For 1990, the time x equals 0, and the initial number of cellular telephone subscribers y is 5,283,000. Thus the y-intercept, and the value of a, is 5,283,000. For 2000, the time x equals 2000 – 1990 or 10, and the number of cellular telephone subscribers is 109,478,000. Example 1-3a

Replace x with 10, y with 109,478,000 and a with 5,283,000. Substitute these values and the value of a into an exponential function to approximate the value of b. Exponential function Replace x with 10, y with 109,478,000 and a with 5,283,000. Divide each side by 5,283,000. Take the 10th root of each side. Example 1-3b

To find the 10th root of 20. 72, use selection To find the 10th root of 20.72, use selection under the MATH menu on the TI-83 Plus. ENTER MATH Keystrokes: 10 5 20.72 1.354063324 Answer: An equation that models the number of cellular telephone subscribers in the U.S. from 1990 to 2000 is Example 1-3c

For 2010, the time x equals 2010 – 1990 or 20. Suppose the number of telephone subscribers continues to increase at the same rate. Estimate the number of US subscribers in 2010. For 2010, the time x equals 2010 – 1990 or 20. Modeling equation Replace x with 20. Use a calculator. Answer: The number of cell phone subscribers will be about 2,136,000,000 in 2010. Example 1-3d

Health In 1991, 4. 9% of Americans had diabetes Health In 1991, 4.9% of Americans had diabetes. By 2000, this percent had risen to 7.3%. a. Write an exponential function of the form could be used to model the percentage of Americans with diabetes. Write the function in terms of x, the number of years since 1991. b. Suppose the percent of Americans with diabetes continues to increase at the same rate. Estimate the percent of Americans with diabetes in 2010. Answer: Answer: 11.4% Example 1-3e

Simplify . Quotient of Powers Answer: Example 1-4a

Simplify . Power of a Power Product of Radicals Answer: Example 1-4b

Simplify each expression. a. b. Answer: Answer: Example 1-4c

Rewrite 256 as 44 so each side has the same base. Solve . Original equation Rewrite 256 as 44 so each side has the same base. Property of Equality for Exponential Functions Add 2 to each side. Divide each side by 9. Answer: The solution is Example 1-5a

Check Original equation Substitute for n. Simplify. Simplify. Example 1-5b

Rewrite 9 as 32 so each side has the same base. Solve . Original equation Rewrite 9 as 32 so each side has the same base. Property of Equality for Exponential Functions Distributive Property Subtract 4x from each side. Answer: The solution is Example 1-5c

Solve each equation. a. b. Answer: Answer: 1 Example 1-5d

Property of Inequality for Exponential Functions Solve Original inequality Rewrite as Property of Inequality for Exponential Functions Subtract 3 from each side. Divide each side by –2. Answer: The solution is Example 1-6a

Check: Test a value of k less than for example, Original inequality Replace k with 0. Simplify. Example 1-6b

Solve Answer: Example 1-6c

End of Lesson 1

Example 1 Logarithmic to Exponential Form Example 2 Exponential to Logarithmic Form Example 3 Evaluate Logarithmic Expressions Example 4 Inverse Property of Exponents and Logarithms Example 5 Solve a Logarithmic Equation Example 6 Solve a Logarithmic Inequality Example 7 Solve Equations with Logarithms on Each Side Example 8 Solve Inequalities with Logarithms on Each Side Lesson 2 Contents

Write in exponential form. Answer: Example 2-1a

Write in exponential form. Answer: Example 2-1b

Write each equation in exponential form. a. b. Answer: Answer: Example 2-1c

Write in logarithmic form. Answer: Example 2-2a

Write in logarithmic form. Answer: Example 2-2b

Write each equation in logarithmic form. a. b. Answer: Answer: Example 2-2c

Let the logarithm equal y. Evaluate Let the logarithm equal y. Definition of logarithm Property of Equality for Exponential Functions Answer: So, Example 2-3a

Evaluate Answer: 3 Example 2-3b

Evaluate . Answer: Example 2-4a

Evaluate . Answer: Example 2-4b

Evaluate each expression. a. b. Answer: 3 Answer: Example 2-4c

Definition of logarithm Solve Original equation Definition of logarithm Power of a Power Simplify. Answer: Example 2-5a

Solve Answer: 9 Example 2-5b

Solve Check your solution. Original inequality Logarithmic to exponential inequality Simplify. Answer: The solution set is Example 2-6a

Check Try 64 to see if it satisfies the inequality. Original inequality Substitute 64 for x. Example 2-6b

Solve Check your solution. Answer: Example 2-6c

Solve Check your solution. Original equation Property of Equality for Logarithmic Functions Subtract 4x and add 3 to each side. Factor. Zero Product Property or Solve each equation. Example 2-7a

Check Substitute each value into the original equation. Substitute 3 for x. Simplify. Original equation Substitute 1 for x. Simplify. Answer: The solutions are 3 and 1. Example 2-7b

Solve Check your solution. Answer: The solutions are 3 and –2. Example 2-7c

Property of Inequality for Logarithmic Functions Solve Original inequality Property of Inequality for Logarithmic Functions Addition and Subtraction Properties of Inequalities We must exclude all values of x such that or Thus the solution set is and This compound inequality simplifies to Answer: The solution set is Example 2-8a

Solve Answer: Example 2-8b

End of Lesson 2

Example 1 Use the Product Property Example 2 Use the Quotient Property Example 3 Use Properties of Logarithms Example 4 Power Property of Logarithms Example 5 Solve Equations Using Properties of Logarithms Lesson 3 Contents

Use to approximate the value of Replace 250 with 53 • 2. Product Property Inverse Property of Exponents and Logarithms Replace with 0.4307. Answer: Thus, is approximately 3.4307. Example 3-1a

Use to approximate the value of Answer: 6.5850 Example 3-1b

Use and to approximate the value of Replace 4 with the quotient Quotient Property and Answer: Thus is approximately 0.7737. Example 3-2a

Use and to approximate the value of Answer: 1.2920 Example 3-2b

Let L1 be the loudness of one lawnmower running. Sound The loudness L of a sound in decibels is given by where R is the sound’s relative intensity. The sound made by a lawnmower has a relative intensity of 109 or 90 decibels. Would the sound of ten lawnmowers running at that same intensity be ten times as loud or 900 decibels? Explain your reasoning. Let L1 be the loudness of one lawnmower running. Let L2 be the loudness of ten lawnmowers running. Example 3-3a

Then the increase in loudness is L2 – L1. Substitute for L1 and L2. Product Property Distributive Property Subtract. Inverse Property of Exponents and Logarithms Example 3-3b

Answer:. No; the sound of ten lawnmowers is perceived Answer: No; the sound of ten lawnmowers is perceived to be only 10 decibels as loud as the sound of one lawnmower, or 100 decibels. Example 3-3c

Sound The loudness L of a sound in decibels is given by Sound The loudness L of a sound in decibels is given by where R is the sound’s relative intensity. The sound made by fireworks has a relative intensity of 1014 or 140 decibels. Would the sound of ten fireworks of that same intensity be ten times as loud or 1400 decibels? Explain your reasoning. Answer: No; the sound of ten fireworks is perceived to be only 10 more decibels as loud as the sound of one firework, or 150 decibels. Example 3-3d

Given that approximate the value of Replace 216 with 63. Power Property Replace with 1.1133. Answer: Example 3-4a

Given that approximate the value of Answer: 5.1700 Example 3-4b

Property of Equality for Logarithmic Functions Solve . Original equation Power Property Quotient Property Property of Equality for Logarithmic Functions Multiply each side by 5. Example 3-5a

Take the 4th root of each side. Answer: Take the 4th root of each side. Example 3-5b

Definition of logarithm Solve . Original equation Product Property Definition of logarithm Subtract 64 from each side. Factor. Zero Product Property or Solve each equation. Example 3-5c

Check Substitute each value into the original equation. Replace x with –4. Since log8 (–4) and log8 (–16) are undefined, –4 is an extraneous solution and must be eliminated. Replace x with 16. Product Property Example 3-5d

Definition of logarithm Answer: The only solution is Example 3-5d

Solve each equation. a. b. Answer: 12 Answer: 8 Example 3-5e

End of Lesson 3

Example 1 Find Common Logarithms Example 2 Solve Logarithmic Equations Using Exponentiation Example 3 Solve Exponential Equations Using Logarithms Example 4 Solve Exponential Inequalities Using Logarithms Example 5 Change of Base Formula Lesson 4 Contents

Use a calculator to evaluate log 6 to four decimal places. ENTER LOG Keystrokes: 6 .7781512503 Answer: about 0.7782 Example 4-1a

Use a calculator to evaluate log 0.35 to four decimal places. ENTER LOG Keystrokes: 0.35 –.4559319557 Answer: about –0.4559 Example 4-1b

Use a calculator to evaluate each expression to four decimal places. a. log 5 b. log 0.62 Answer: 0.6990 Answer: –0.2076 Example 4-1c

Write each side using 10 as a base. Earthquake The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log The San Fernando Valley earthquake of 1994 measured 6.6 on the Richter scale. How much energy did this earthquake release? Write the formula. Replace M with 6.6. Simplify. Write each side using 10 as a base. Example 4-2a

Inverse Property of Exponents and Logarithms Use a calculator. Answer: The amount of energy released was about ergs. Example 4-2b

Earthquake The amount of energy E, in ergs, that an earthquake releases is related to its Richter scale magnitude M by the equation log In 1999 an earthquake in Turkey measured 7.4 on the Richter scale. How much energy did this earthquake release? Answer: about Example 4-2c

Property of Equality for Logarithmic Functions Solve Original equation Property of Equality for Logarithmic Functions Power Property of Logarithms Divide each side by log 62. Use a calculator. Answer: Example 4-3a

Check You can check this answer by using a calculator or by using estimation. Since and the value of x is between 2 and 3. Thus, 2.5643 is a reasonable solution. Example 4-3b

Solve Answer: 2.5789 Example 4-3c

Solve Original inequality Property of Inequality for Logarithmic Functions Power Property of Logarithms Distributive Property Subtract 5x log 3 from each side. Example 4-4a

Use a calculator. Simplify. Distributive Property Divide each side by Switch > to < because is negative. Use a calculator. Simplify. Example 4-4b

Negative Exponent Property Check: Original inequality Replace x with 0. Simplify. Negative Exponent Property Answer: The solution set is Example 4-4d

Solve Answer: Example 4-4e

Answer: The value of is approximately 2.6309. Express in terms of common logarithms. Then approximate its value to four decimal places. Change of Base Formula Use a calculator. Answer: The value of is approximately 2.6309. Example 4-5a

Express. in terms of common logarithms Express in terms of common logarithms. Then approximate its value to four decimal places. Answer: Example 4-5b

End of Lesson 4

Example 1 Evaluate Natural Base Expressions Example 2 Evaluate Natural Logarithmic Expressions Example 3 Write Equivalent Expressions Example 4 Inverse Property of Base e and Natural Logarithms Example 5 Solve Base e Equations Example 6 Solve Base e Inequalities Example 7 Solve Natural Log Equations and Inequalities Lesson 5 Contents

Use a calculator to evaluate to four decimal places. ENTER 2nd Keystrokes: [ex] 0.5 1.648721271 Answer: about 1.6487 Example 5-1a

Use a calculator to evaluate to four decimal places. ENTER 2nd Keystrokes: [ex] –8 .0003354626 Answer: about 0.0003 Example 5-1b

Use a calculator to evaluate each expression to four decimal places. b. Answer: 1.3499 Answer: 0.1353 Example 5-1c

Use a calculator to evaluate In 3 to four decimal places. Keystrokes: ENTER LN 3 1.098612289 Answer: about 1.0986 Example 5-2d

Use a calculator to evaluate In to four decimal places. Keystrokes: ENTER LN 1 ÷ 4 –1.386294361 Answer: about –1.3863 Example 5-2e

Use a calculator to evaluate each expression to four decimal places. a. In 2 b. In Answer: 0.6931 Answer: –0.6931 Example 5-2f

Write an equivalent logarithmic equation for . Answer: Example 5-3a

Write an equivalent exponential equation for Answer: Example 5-3b

Write an equivalent exponential or logarithmic equation. a. b. Answer: Answer: Example 5-3c

Evaluate Answer: Example 5-4a

Evaluate . Answer: Example 5-4b

Evaluate each expression. a. b. Answer: 7 Answer: Example 5-4c

Subtract 4 from each side. Solve Original equation Subtract 4 from each side. Divide each side by 3. Property of Equality for Logarithms Inverse Property of Exponents and Logarithms Divide each side by –2. Use a calculator. Answer: The solution is about –0.3466. Example 5-5a

Check You can check this value by substituting –0 Check You can check this value by substituting –0.3466 into the original equation or by finding the intersection of the graphs of and Example 5-5b

Solve Answer: 0.8047 Example 5-5c

What is the balance after 8 years? Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. What is the balance after 8 years? Continuous compounding formula Replace P with 700, r with 0.06, and t with 8. Simplify. Use a calculator. Answer: The balance after 8 years would be $1131.25. Example 5-6a

How long will it take for the balance in your account to reach at least $2000? The balance is at least $2000. 2000 Replace A with 700e(0.06)t. A  Write an inequality. Divide each side by 700. Property of Inequality for Logarithms Inverse Property of Exponents and Logarithms Example 5-6b

Divide each side by 0.06. Use a calculator. Answer: It will take at least 17.5 years for the balance to reach $2000. Example 5-6c

a. What is the balance after 7 years? Savings Suppose you deposit $700 into an account paying 6% annual interest, compounded continuously. a. What is the balance after 7 years? b. How long will it take for the balance in your account to reach at least $2500? Answer: $1065.37 Answer: at least 21.22 years Example 5-6d

Write each side using exponents and base e. Solve Original equation Write each side using exponents and base e. Inverse Property of Exponents and Logarithms Divide each side by 3. Use a calculator. Answer: The solution is 0.5496. Check this solution using substitution or graphing. Example 5-7a

Write each side using exponents and base e. Solve Original inequality Write each side using exponents and base e. Inverse Property of Exponents and Logarithms Add 3 to each. Divide each side by 2. Use a calculator. Example 5-7b

Answer: The solution is all numbers less than. 7 Answer: The solution is all numbers less than 7.5912 and greater than 1.5. Check this solution using substitution. Example 5-7c

Solve each equation or inequality. a. b. Answer: about 1.0069 Answer: Example 5-7d

End of Lesson 5

Example 1 Exponential Decay of the Form y = a(1 – r)t Example 2 Exponential Decay of the Form y = ae–kt Example 3 Exponential Growth of the Form y = a(1 + r )t Example 4 Exponential Growth of the Form y = aekt Lesson 6 Contents

Caffeine A cup of coffee contains 130 milligrams of caffeine Caffeine A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for 90% of this caffeine to be eliminated from a person’s body? Explore The problem gives the amount of caffeine consumed and the rate at which the caffeine is eliminated. It asks you to find the time it will take for 90% of the caffeine to be eliminated from a person’s body. Use the formula Let t be the number of hours since drinking the coffee. The amount remaining y is 10% of 130 or 13. Plan Example 6-1a

Exponential decay formula Solve Exponential decay formula Replace y with 13, a with 130, and r with 0.11. Divide each side by 130. Property of Equality for Logarithms Power Property for Logarithms Divide each side by log 0.89. Use a calculator. Example 6-1b

Exponential decay formula Answer: It will take approximately 20 hours for 90% of the caffeine to be eliminated from a person’s body. Examine Use the formula to find how much of the original 130 milligrams of caffeine would remain after 20 hours. Exponential decay formula Replace a with 130, r with 0.11 and t with 20. Ten percent of 130 is 13, so the answer seems reasonable. Example 6-1c

Caffeine A cup of coffee contains 130 milligrams of caffeine Caffeine A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for 80% of this caffeine to be eliminated from a person’s body? Answer: 13.8 hours Example 6-1d

Exponential decay formula Geology The half-life of Sodium-22 is 2.6 years. What is the value of k for Sodium-22? Exponential decay formula Replace y with 0.5a and t with 2.6. Divide each side by a. Property of Equality for Logarithmic Functions Inverse Property of Exponents and Logarithms Divide each side by –2.6. Example 6-2a

Use a calculator. Answer: The constant k for Sodium-22 is 0.2666. Thus, the equation for the decay of Sodium-22 is where t is given in years. Example 6-2b

Formula for the decay of Sodium-22 A geologist examining a meteorite estimates that it contains only about 10% as much Sodium-22 as it would have contained when it reached the surface of the Earth. How long ago did the meteorite reach the surface of the Earth? Formula for the decay of Sodium-22 Replace y with 0.1a. Divide each side by a. Property of Equality for Logarithms Example 6-2c

Inverse Property for Exponents and Logarithms Divide each side by –0.2666. Use a calculator. Answer: It was formed about 9 years ago. Example 6-2d

a. What is the value of k for radioactive iodine? Health The half-life of radioactive iodine used in medical studies is 8 hours. a. What is the value of k for radioactive iodine? b. A doctor wants to know when the amount of radioactive iodine in a patient’s body is 20% of the original amount. When will this occur? Answer: Answer: about 19 hours later Example 6-2e

Multiple-Choice Test Item The population of a city of one million is increasing at a rate of 3% per year. If the population continues to grow at this rate, in how many years will the population have doubled? A 4 years B 5 years C 20 years D 23 years Read the Test Item You want to know when the population has doubled or is 2 million. Use the formula Example 6-3a

Exponential growth formula Solve the Test Item Exponential growth formula Replace y with 2,000,000, a with 1,000,000, and r with 0.03. Divide each side by 1,000,000. Property of Equality for Logarithms Power Property of Logarithms Example 6-3b

Divide each side by ln 1.03. Use a calculator. Answer: D Example 6-3c

Multiple-Choice Test Item The population of a city of 10,000 is increasing at a rate of 5% per year. If the population continues to grow at this rate, in how many years will the population have doubled? A 10 years B 12 years C 14 years D 18 years Answer: C Example 6-3d

You want to find t such that Population As of 2000, Nigeria had an estimated population of 127 million people and the United States had an estimated population of 278 million people. The growth of the populations of Nigeria and the United States can be modeled by and , respectively. According to these models, when will Nigeria’s population be more than the population of the United States? You want to find t such that Replace N(t) with and U(t) with Example 6-4a

Property of Inequality for Logarithms Product Property of Logarithms Inverse Property of Exponents and Logarithms Subtract ln 278 and 0.026t from each side. Divide each side by –0.017. Use a calculator. Answer: After 46 years or in 2046, Nigeria’s population will be greater than the population of the U.S. Example 6-4b

Answer: after 109 years or in the year 2109 Population As of 2000, Saudi Arabia had an estimated population of 20.7 million people and the United States had an estimated population of 278 million people. The growth of the populations of Saudi Arabia and the United States can be modeled by and , respectively. According to these models, when will Saudi Arabia’s population be more than the population of the United States? Answer: after 109 years or in the year 2109 Example 6-4c

End of Lesson 6

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