<p> Chapter 7: Exponential and Logarithmic Functions</p><p>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</p><p>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</p><p>Examples: 1. Graph the functions. State the domain and range of each.</p><p> a. y = 5x b. c. d. </p><p>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?</p><p> 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</p><p>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</p><p>Examples: 1. Graph. State the domain and range of each. a. b. </p><p> c. d. </p><p>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.</p><p> b. Graph the model.</p><p>7.3 Use Functions Involving e natural base e: as n approaches +∞, , e ; the natural base e is irrational </p><p>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</p><p>Examples: 1. Simplify the expression. a. e9 · e6 b. c. </p><p>2. Use a calculator to evaluate the expression to the nearest thousandth. a. e6 b. e-0.28</p><p>3. Graph the function. State the domain and range.</p><p> 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.</p><p>5. You deposit $3000 in an account that pays 3.5% annual interest compounded continuously. What is the balance after 3 years?</p><p>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)</p><p> common logarithm: common logarithm with base 10 (denoted by log10 x or log x)</p><p> natural logarithm: a logarithm with base e (denoted by log e x or ln x)</p><p>Parent Graphs for Logarithmic Functions</p><p>Put Graph Put Graph Here Here</p><p>Examples: 1. Rewrite the equation in exponential form. a. b. c. d. </p><p>2. Evaluate the logarithm. a. b. c. d. </p><p>3. Use a calculator to evaluate the logarithm. a. log 0.85 b. ln 22</p><p>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.</p><p>5. Simplify the expression. ln 9 x a. e b. log 3 27</p><p>6. Find the inverse of the function. a. y = 8 x b. y = ln (x – 4)</p><p>7. Graph the function. a. log 2 x b. </p><p>8. Graph the y = log 3 (x – 2) + 4. State the domain and range.</p><p>7.5 Apply Properties of Logarithms</p><p>Properties of Logarithms: Let b, m, and n be positive numbers such that b ≠ 1.</p><p>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</p><p>Change-of-Base Formula If a, b, and c are positive numbers with b ≠ 1 and c ≠ 1, then: ; in particular , and </p><p>Examples:</p><p>1. Use log 3 12 2.262 and log 3 2 0.631 to evaluate the logarithm.</p><p> a. log 3 6 b. log 3 24 c. log 3 32 </p><p>2. Expand the .</p><p>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</p><p>4. Evaluate log 6 24 using common logarithms and natural logarithms.</p><p>5. The Richter scale is used to measure the magnitude of earthquakes. If an earthquake has intensity I, then </p><p> 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?</p><p>7.6 Solving Exponential and Logarithmic Equations</p><p>Property of Equality for Exponential Equations if b is a positive number other than 1, then bx = by if an only if x = y</p><p>Property of Equality for Logarithmic Equations</p><p> if b, x, and y are positive numbers with b ≠ 1, then log b x = log b y if and only if x = y </p><p>Examples: 1. Solve. x x a. 3 = b. 9 = 35 c. log 4 (2x + 8) = log 4 (6x – 12)</p><p> 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</p><p>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?</p><p>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?</p><p>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</p><p>Examples: 1. Write an exponential function y = abx whose graph passes through (1, 10) and (4, 80).</p><p>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</p><p> 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)?</p><p> b. Find an exponential model for the original data.</p><p>3. Use a graphing calculator to find an exponential model for the data in Example 2. Predict the enrollment for the sixth year.</p><p>4. Write an exponential function y = axb whose graph passes through (4, 6) and (8, 15).</p><p>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 </p><p> 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.</p><p> 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)?</p><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.</p>