Chapter 7: Exponential and Logarithmic Functions Big ideas:  Graphing Exponential and Logarithmic Functions  Solving exponential and logarithmic equations  Writing and applying exponential and power functions

BELL RINGER  Evaluate the expression:  5 -2  -3 * 4 3/2

Lesson 1: Graph Exponential Growth Functions (abbreviated)

Essential question What is the equation to find compounded interest?

VOCABULARY  Exponential function: A function of the form y=ab x, where a≠ 0, b>0, and b≠1  Exponential growth function: If a>0 and b<1, then the function y=ab x is an exponential growth function with growth factor b  Growth factor: The quantity b in the exponential growth function y=ab x with a>0 and b<1  Asymptote: A line that a graph approaches more and more closely

EXAMPLE 5 Find the balance in an account = = 4000(1.0073) 4 = P = 4000, r = , n = 4, t = 1 Simplify. Use a calculator. ANSWER The balance at the end of 1 year is $ SOLUTION a. With interest compounded quarterly, the balance after 1 year is: A = P 1 + r n nt Write compound interest formula.

EXAMPLE 5 Find the balance in an account b. With interest compounded daily, the balance after 1 year is: A = P 1 + r n nt = = 4000( ) 365 = Write compound interest formula. P = 4000, r = , n = 365, t = 1 Simplify. Use a calculator. ANSWER The balance at the end of 1 year is $

GUIDED PRACTICE for Example 5 6. FINANCE You deposit $2000 in an account that pays 4% annual interest. Find the balance after 3 years if the interest is compounded daily. $ ANSWER a. With interest compounded daily, the balance after 3 years is:

Essential question

BELL RINGER

Lesson 3: Use Functions Involving e

Essential question When is the natural base e useful?

VOCABULARY

EXAMPLE 1 Simplify natural base expressions Simplify the expression. a.e2e2 e5e5 = e = e7e7 b. 12e4e4 3e3e3 = e 4 – 3 4 = 4e4e (5(5 ) c.e –3x 2 = 5252 ( ) 2 = 25 e –6x = 25 e6xe6x

EXAMPLE 2 Evaluate natural base expressions Use a calculator to evaluate the expression. a.e4e4 b.e –0.09 ExpressionKeystrokesDisplay [ ]exex 4 exex 0.09

GUIDED PRACTICE for Examples 1 and 2 Simplify the expression. 1.e7e7 e4e4 e e8e8 4e5e5 12 e2e e –3 6e56e5 6e3e3 SOLUTION

GUIDED PRACTICE for Examples 1 and 2 Simplify the expression. 4. ( 10 e –4x ) e 12x Use a calculator to evaluate 5.e 3/ SOLUTION

Essential question When is the natural base e useful? This is a special irrational number, useful in many applications of exponential functions (continuously increasing or decreasing biological phenomena and continuously compounded interest)

BELL RINGER  Find the inverse of the function y = 3x-5

Lesson 4: Evaluate Logarithms

Essential question What is the relationship between exponential and logarithmic functions?

VOCABULARY  Exponential Equation: An equation in which a variable expression occurs as an exponent  Logarithmic Equation: An equation that involves a logarithm of a variable expression  Extraneous Solution: An apparent solution that must be rejected because it does not satisfy the original equation.

EXAMPLE 1 Rewrite logarithmic equations Logarithmic FormExponential Form 2323 = 8 a. = 2 log = 1b. 4 log 1 = 0 = c. 12 log 121 = d. 1/4 log – = 12 4 = –1 1 4

GUIDED PRACTICE for Example 1 Rewrite the equation in exponential form. Logarithmic FormExponential Form 3434 = = 3 log = log 7 = 1 = log 10 = 4. 1/2 log – = 1 32 = –5 1 2

EXAMPLE 2 Evaluate logarithms 4 log a.64 b. 5 log 0.2 Evaluate the logarithm. b log To help you find the value of y, ask yourself what power of b gives you y. SOLUTION 4 to what power gives 64 ? a. 4 log , so = 3.3. = 64 5 to what power gives 0.2 ? b. = 5 –1  0.2, so – log =

EXAMPLE 2 Evaluate logarithms Evaluate the logarithm. b log To help you find the value of y, ask yourself what power of b gives you y. SOLUTION = – , so 1/5 log 125 = –3. c. to what power gives 125 ? 1 5 d.36 to what power gives 6 ? 36 1/2 6, so 36 log 6 == 1 2. d. 36 log 6 c. 1/5 log 125

EXAMPLE 3 Evaluate common and natural logarithms ExpressionKeystrokesDisplay a. log 8 b. ln 0.3 Check – e –1.204

GUIDED PRACTICE for Examples 2, 3 and 4 Evaluate the logarithm. Use a calculator if necessary log SOLUTION 27 log 6.3 SOLUTION 7. log ln –0.288 SOLUTION

EXAMPLE 5 Use inverse properties Simplify the expression. a. 10 log 4 b. 5 log 25 x SOLUTION Express 25 as a power with base 5. a.10 log 4 = 4 b. 5 log 25 x = ( 5252 ) x 5 log = 5 52x52x 2x2x = Power of a power property b log x b = x b log bxbx = x

EXAMPLE 6 Find inverse functions Find the inverse of the function. SOLUTION b. a. y = 6 x b. y = ln (x + 3) a. 6 log From the definition of logarithm, the inverse of y = 6 x is y = x.x. Write original function. y = ln (x + 3) Switch x and y. x = ln (y + 3) Write in exponential form. Solve for y. = exex (y + 3) = e x – 3 y ANSWER The inverse of y = ln (x + 3) is y =e x – 3.

GUIDED PRACTICE for Examples 5 and 6 Simplify the expression. SOLUTION log x x log 7 –3x SOLUTION –3x

GUIDED PRACTICE for Examples 5 and 6 Simplify the expression. SOLUTION log 64 x 6x6x 13.e ln20 SOLUTION 20

GUIDED PRACTICE for Examples 5 and 6 Find the inverse of 14. y = 4 x SOLUTION 4 log y = x.x. y = ln (x – 5). Find the inverse of 15. y =e x + 5. SOLUTION

Essential question What is the relationship between exponential and logarithmic functions? Exponential and logarithmic functions with the same base are inverses

BELL RINGER  Evaluate:  Log  Log 32 2

Lesson 5: Apply Properties of Logarithms

Essential question How can you use a calculator to evaluate a logarithm when the base is not 10 or e?

VOCABULARY  Base: The number or expression that is used as a factor in a repeated multiplication

EXAMPLE 1 Use properties of logarithms b. 4 log 21 a. 4 log 3 7 = 3 – = –0.612 = 4 log (3 7) = 3 4 log = – log Use and 4 log to evaluate the logarithm. Quotient property Simplify. Use the given values of 3 4 log and Write 21 as 3 7. Product property Use the given values of 3 4 log and Simplify.

EXAMPLE 1 Use properties of logarithms c. 4 log log Use and 4 log to evaluate the logarithm = 4 log = (1.404) Write 49 as 7272 Power property Use the given value of log Simplify. 4 log = 27

GUIDED PRACTICE for Example log 40 – log 5 6 Use and to evaluate the logarithm. 6 log log SOLUTION

EXAMPLE 2 Expand a logarithmic expression Power property Expand 6 log 5x35x3 y 6 5x35x3 y Quotient property = 5 6 log x3x3 y 6 – 6 + = 5 6 x y 6 – Product property SOLUTION = 5x35x3 y 6 log – 6

GUIDED PRACTICE for Examples 2 and 3 Expand 5. log 3x4x4. SOLUTION log log x SOLUTION ln 9 Condense ln ln 3 – ln

EXAMPLE 4 Use the change-of-base formula SOLUTION 3 log 8 Evaluate using common logarithms and natural logarithms. Using common logarithms: Using natural logarithms: 3 log 8 = log 8 log log 8 = ln 8 ln

GUIDED PRACTICE for Examples 4 and 5 Use the change-of-base formula to evaluate the logarithm. 5 log 87. SOLUTION about log 148. SOLUTION about 1.269

GUIDED PRACTICE for Examples 4 and 5 Use the change-of-base formula to evaluate the logarithm. 26 log 99. SOLUTION about about log 30

Essential question How can you use a calculator to evaluate a logarithm when the base is not 10 or e? The change-of-base formula allows you to evaluate a logarithm with any base by finding the quotient of two common logarithms or of two natural logarithms. You can rewrite the logarithm using the change of base formula then evaluate the resulting expression using a calculator.