Chapter 7: Exponential and Logarithmic Functions

7.1 Graph Exponential Growth Functions exponential function: y = abx where a ≠ 0 and b is a positive number other than 1 exponential growth function: y = abx where a > 0 and b > 1 (b is the growth factor) asymptote: a line that a graph approaches but never actually reaches

Compound Interest: , A = amount in the account after t years, P = initial principal, r = annual rate, n = the number of times it is compounded per year

Examples: 1. Graph the functions. State the domain and range of each.

a. y = 5x b. c. d.

2. In 1970, the population of Kern County, California, where Bakersfield is located, was about 330,000. From 1970 to 2000, the county population grew at an average annual rate of about 2.4%. a. Write an exponential growth model giving the population P of Kern County t years after 1970. About how many people lived in Kern County in 1990?

b. Graph the model. 6. You deposit $5500 in an account that pays 3.6% annual interest. Find the balance after 2 years if interest is compounded with the given frequency: a. semiannually b. monthly

7.2 Graph Exponential Decay Functions exponential decay function: y = abx where a > 0 and 0 < b < 1 (b is the decay factor) exponential decay model: y = a(1 – r)t, where t is the number of years, a is the initial amount, and r is the percent decrease expressed as a decimal

Examples: 1. Graph. State the domain and range of each. a. b.

2. A new car costs $25,000. The value of the car decreases by 15% each year. a. Write an exponential decay model giving the car’s value y (in dollars) after t years. Estimate the value after 4 years.

b. Graph the model.

7.3 Use Functions Involving e natural base e: as n approaches +∞, , e ; the natural base e is irrational

Continuously Compounded Interest: A = Pert, where A is the amount in the account after t years, P represents the principal amount invested, r is the annual interest rate

Examples: 1. Simplify the expression. a. e9 · e6 b. c.

2. Use a calculator to evaluate the expression to the nearest thousandth. a. e6 b. e-0.28

3. Graph the function. State the domain and range.

a. y = 4e0.5x b. y = e-1.5(x + 2) – 4 4. Annual sales of a certain product can be modeled by the function , where S is the number of years since the product went on the market. a. Graph the model. b. Use the graph to estimate the annual sales 6 years after the product went on the market.

5. You deposit $3000 in an account that pays 3.5% annual interest compounded continuously. What is the balance after 3 years?

7.4 Evaluate Logarithms and Graph Logarithmic Functions Definition of Logarithm with Base b: let b and y be positive numbers with b ≠ 1; the logarithm of y with base b x is denoted by log b y = x if and only if b = y (read log base b of y)

common logarithm: common logarithm with base 10 (denoted by log10 x or log x)

natural logarithm: a logarithm with base e (denoted by log e x or ln x)

Parent Graphs for Logarithmic Functions

Put Graph Put Graph Here Here

Examples: 1. Rewrite the equation in exponential form. a. b. c. d.

2. Evaluate the logarithm. a. b. c. d.

3. Use a calculator to evaluate the logarithm. a. log 0.85 b. ln 22

4. The sales of a certain video game can be modeled by y = 20 ln (x – 1) + 35, where y is the monthly number (in thousands) of games sold the xth month after the game is released for sale (x > 1). Estimate the number of video games sold during the 10th month after the game is released.

5. Simplify the expression. ln 9 x a. e b. log 3 27

6. Find the inverse of the function. a. y = 8 x b. y = ln (x – 4)

7. Graph the function. a. log 2 x b.

8. Graph the y = log 3 (x – 2) + 4. State the domain and range.

7.5 Apply Properties of Logarithms

Properties of Logarithms: Let b, m, and n be positive numbers such that b ≠ 1.

Product Property: log b mn = log b m + log b n Quotient Property: = log b m – log b n n Power Property: log b m = n log b m

Change-of-Base Formula If a, b, and c are positive numbers with b ≠ 1 and c ≠ 1, then: ; in particular , and

Examples:

1. Use log 3 12 2.262 and log 3 2 0.631 to evaluate the logarithm.

a. log 3 6 b. log 3 24 c. log 3 32

2. Expand the .

3. Which of the following is equivalent to ln 8 + 2 ln 5 – ln 10? A. ln 4 B. ln 8 C. ln 18 D. ln 20

4. Evaluate log 6 24 using common logarithms and natural logarithms.

5. The Richter scale is used to measure the magnitude of earthquakes. If an earthquake has intensity I, then

its magnitude on the Richter scale, R, is given by the function R(I) = , where IO is the intensity of a barely felt earthquake. If the intensity of a barely felt earthquake is 50 times that of another, how many points greater is the bigger earthquake on the Richter scale?

7.6 Solving Exponential and Logarithmic Equations

Property of Equality for Exponential Equations if b is a positive number other than 1, then bx = by if an only if x = y

Property of Equality for Logarithmic Equations

if b, x, and y are positive numbers with b ≠ 1, then log b x = log b y if and only if x = y

Examples: 1. Solve. x x a. 3 = b. 9 = 35 c. log 4 (2x + 8) = log 4 (6x – 12)

d. log 7 ( 3x – 2) = 2 2. What is (are) the solution(s) of log 6 3x + log 6 (x – 4) = 2? A. – 6, 2 B. – 2, 6 C. 2 D. 6

3. The population of deer in a forest preserve can be modeled by the equation P = 50 + 200 ln (t + 1), where t is the time in years from the present. In how many years will the deer population reach 500?

4. hot chocolate that has been heated to 900C is poured into a mug and placed on a table in a room with a temperature of 200C. If r = 0.145 when the time t is measured in minutes, how long will it take for the hot chocolate to cool to a temperature 300C?

7.7 Write and Apply Exponential and Power Functions power function: has the form y = abx where a is a real number and b is a rational number exponential function: y = abx where a ≠ 0 and b is a positive number other than 1

Examples: 1. Write an exponential function y = abx whose graph passes through (1, 10) and (4, 80).

2. The table shows the number y of students enrolled in an elementary school during the xth year that the school as been open. x 1 2 3 4 5 y 370 417 460 523 598

a. Draw a scatter plot of the data pairs (x, ln y). Is an exponential model a good fit for the original data pairs (x, y)?

b. Find an exponential model for the original data.

3. Use a graphing calculator to find an exponential model for the data in Example 2. Predict the enrollment for the sixth year.

4. Write an exponential function y = axb whose graph passes through (4, 6) and (8, 15).

Planet R P Mercury 0.39 0.24 Venus 0.72 0.62 Earth 1.00 1.00 Mars 1.52 1.88 Jupiter 5.20 11.9 5. The period of a planet is the length of time it takes for the planet to make one complete revolution around

the Sun. The table shows the mean distance R1, in astronomical units, of each several planets from the Sun and the period P, in years of each of these planets.

a. Draw a scatter plot of the data pairs (ln R, ln P). Is an exponential model a good fit for the original pairs (R, P)?

b. Find a power model for the original data. 6. Use a graphing calculator to find a power model for the data in Example 5. Estimate the period of Saturn, whose mean distance from the Sun is 9.54 astronomical units.