Name Date Per. Ms. Williams/Mrs. Hertel

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1 Name Date Per. Ms. Williams/Mrs. Hertel Day 7: Solving Exponential Word Problems involving Logarithms Warm Up Exponential growth occurs when a quantity increases by the same rate r in each period t. When this happens, the value of the quantity at any given time can be calculated as a function of the rate and the original amount. Exponential decay occurs when a quantity decreases by the same rate r in each time period t. Just like exponential growth, the value of the quantity at any given time can be calculated by using the rate and the original amount. In Summary,

2 Example 1: Growth The original value of a painting is $9,000 and the value increases by 7% each year. Part a: Then find the painting s value in 15 years. Part b: In what year, will the painting be worth $50,000? Example 2: Decay The population of a town is decreasing at a rate of 3% per year. In 2000 there were 1700 people. Part a: Find the population in Part b: In what year, will the population be double?

3 3) Is the equation A = 3200 (0.70) t a model of exponential growth or exponential decay, and what is the rate (percent) of change per time period? Explain here! 1) exponential growth and 30% 2) exponential growth and 70% 3) exponential decay and 30% 4) exponential decay and 70% 4) Is the equation A = 1756 (1.17) t a model of exponential growth or exponential decay, and what is the rate (percent) of change per time period? 1) exponential growth and 17% Explain here! 2) exponential growth and 83% 3) exponential decay and 17% 4) exponential decay and 83% 5)

4 Graphs of Logarithmic Functions Using the table below: a) Complete the table of values for y= 2 x b) sketch the graph of y= 2 x x y ) Recall: How do we find the inverse of a function? Properties of Domain: Properties of Domain: Range: Range: Find the inverse algebraically. Asymptote: Asymptote: x-intercept: x-intercept: y-intercept: y-intercept: 3) Graph the inverse of the function y = 2 x.

5 Rule for Graphing Exponential Functions Rule for Graphing Log Functions x y x y -1 1 b 1 b b 1 0 b 1

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10 Word Problems Homework Day 1 Write an exponential growth/ decay function to model each situation. Then find the value of the function after the given number of years. 1) 2) 3) 4) Is the equation A = 10,000 (0.45) t a model of exponential growth or exponential decay, and what is the rate (percent) of change per time period? 1) exponential growth and 45% 2) exponential growth and 55% Explain here! 3) exponential decay and 45% 4) exponential decay and 55% 5) Is the equation A = 5400 (1.07) t a model of exponential growth or exponential decay, and what is the rate (percent) of change per time period? 1) exponential growth and 7% 2) exponential growth and 93% Explain here! 3) exponential decay and 7% 4) exponential decay and 93%

11 6) 7) Sketch below the graph of. Then, state the domain and range of the graph. Write the equation of the asymptote. 8)

12 Name Date Per. Ms. Williams/Mrs. Hertel Day 8: Solving Exponential Word Problems involving Logarithms Warm Up 1) In January 1995, the population of a small town was 8,000 people. Each year after 1995, the population decreased by 1%. a. Find the population of the town in January b. If this rate of decrease continues unchanged, what is the expected population of the town in January 2010? 2)

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14 Example 2: Level B

15 Example 3: Level A Example 4: Level B

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17 Summary Exit Ticket

18 Day 8 Homework 1) 2) 3)

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21 NAME: Algebra 2/Trig Unit 10: Logarithms REVIEW SHEET DATE: PERIOD: Converting and Solving Logarithms 1. Solve for x: log 3 (x - 1) = 2 5. Solve for x to the nearest hundredth: 2. Find the value of to four decimal places. Using the Power Law 6. Solve for x to the nearest thousandth: 3. The relationship between the relative size of an earthquake, S, and the measure of the earthquake on the Richter scale, R, is given by the equation log S = R. If an earthquake measured 6.2 on the Richter scale, what was its relative size to the nearest tenth? 7. Using logarithms, find w to the nearest tenthousandth: 4. The expression is equivalent to 1) 8 2) 2 3) 4)

22 Product and Quotient Laws 8. The expression is equivalent to 1) 3) 2) 4) Substitution with Logarithms 10. If and, what is? (1) x 2 y (3) x y 2 (2) 2x 2y y (4) x 2 9. The expression ( 1) 2) 3) 4) ) is equivalent to 11. Given: and Express in terms of p and q: Solving Logarithmic Equations 12. Solve algebraically for all values of x:

23 13. Solve for x: Undefined Logarithms 14. The expression log (x 2-4) is defined for all values of x such that (1) -2 x 2 (3) x 2 or x -2 (2) -2 < x < 2 (4) x > 2 or x < -2 Solving Logarithmic Word Problems 15. The scientists in a laboratory company raise amebas to sell to schools for use in biology classes. They know that one ameba divides into two amebas every hour and that the formula t = log 3 N can be used to determine how long in hours, t, it takes to produce a certain number of amebas, N. Determine, to the nearest hundredth of an hour, how long it takes to produce 5,000 amebas if they start with one ameba.

24 16. Sean invests $10,000 at an annual rate of 5% compounded continuously, according to the formula A Pe rt, where A is the amount, P is the principal, r is the rate of interest, and t is time, in years. Determine, to the nearest dollar, the amount of money he will have after 2 years. Determine how many years, to the nearest year, it will take for his initial investment to double. Inverse and Graphs of Logarithms 17. What is the inverse of the function y = log 3 x (1) 3 y = x (3) x 3 = y (2) 3 x = y (4) y = x Graph the equations on the same set of axes. State the domain and range of. Write the equation of the asymptote of.