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Welcome to Interactive Chalkboard Algebra 2 Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Send all inquiries to: GLENCOE DIVISION Glencoe/McGraw-Hill 8787 Orion Place Columbus, Ohio 43240

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Splash Screen

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End of Lesson 4

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

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Example 5-1a Answer: about 1.6487 ENTER2nd Keystrokes: [e x ] 0.5 1.648721271 Use a calculator to evaluate to four decimal places.

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Example 5-1b Answer: about 0.0003 ENTER2nd Keystrokes: [e x ] –8.0003354626 Use a calculator to evaluate to four decimal places.

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

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

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

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

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

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

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

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Example 5-4a Evaluate Answer:

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Example 5-4b Evaluate. Answer:

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

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

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

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Example 5-5c Answer: 0.8047 Solve

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

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

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

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Example 5-6d 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

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

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

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

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

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End of Lesson 5

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

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Example 6-1a 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

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Example 6-1b 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.

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Example 6-1c 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. Ten percent of 130 is 13, so the answer seems reasonable. Exponential decay formula Replace a with 130, r with 0.11 and t with 20.

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Example 6-1d 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

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Example 6-2a Geology The half-life of Sodium-22 is 2.6 years. What is the value of k for Sodium-22? 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. Exponential decay formula

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Example 6-2b 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.

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Example 6-2c 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

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

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Example 6-2e 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: about 19 hours later Answer:

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You want to know when the population has doubled or is 2 million. Use the formula Example 6-3a 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 yearsB 5 years C 20 yearsD 23 years Read the Test Item

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Example 6-3b 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

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

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Example 6-3d 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 yearsB 12 years C 14 yearsD 18 years Answer: C

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Example 6-4a 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? Replace N(t) with and U(t) with

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

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Example 6-4c 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?

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