Chapter 8 Eponential and Logarithmic Functions
Lesson 8-1 Eploring Eponential Models
Eponential Function The general form of an eponential function is y = ab. Growth Factor When the value of b is greater than 1 Decay Factor When the value of b is less than 1 When you see words like Increase or appreciation, think growth Decrease or depreciation, think decay
Eample 4 Page 427, #16, 20 Without graphing determine whether each function represents eponential growth or eponential decay. y 129(1.63) y 5 4 6 Eponential Growth Eponential Decay
Eample 1 Page 426, #4 Graph each Function. y 9(3) Eponential Growth y 0 9 1 3 2 1 6 0.0123
Eample 5 Page 427, #26 Graph each Function. Eponential Decay y 0.25 y 1 4 0.5 0.5 1 0.25
Asymptote An asymptote is a line that a graph approaches as or y increases in absolute value.
Eample 2 A Carl s weight at 12 yr is 82 lb. Assume that his weight increases at a rate of 16% each year. Write an eponential function to model the increase. Calculate his weight after 5 yr. Step 1 Find a and b. y ab a is the original a 82 b is the growth factor or decay factor b 1r 1 0.16 1.16
Eample 2 A Step 2 Find the eponential function. y ab y 82(1.16) a b 82 1.16 Step 3 Calculate his weight after 5 yr. 5 y 82(1.16) 172.228 If the model is correct, Carl will weight about 172 lb in 5 yr.
Eample 2 B A motorcycle purchased for $9000 today it will be worth 6% less each year. For what can you epect to sell the motorcycle at the end of 5 yr? Step 1 Find a and b. y ab a is the original a 9000 b is the growth factor or decay factor b 1r 1 0.06 0.94
Eample 2 B Step 2 Find the eponential function. y ab y 9000(0.94) a b 9000 0.94 Step 3 Calculate the sale price after 5 yr. 5 y 9000(0.94) $6605.14
Lesson 8-2 Properties of Eponential Functions
Eample 1 Page 434, #2 Graph each function. Label the asymptote of each graph y 1 2 y 0 1 1 0.50 2 0.25 3 0.125 asymptote is y = 0
Eample 3 Page 434, #16 Iodine-131 is used to find leaks in water pipes. It has a half-life of 8.14 days. Write the eponential decay function for a 200-mg sample. Find the amount of iodine-131 remaining after 72 days. Step 1 Find a and b. y ab a is the original a 200 b is ½ and is the number of days 1 b 2 8.14
Eample 3 Page 434, #16 Step 2 Find the eponential function. y ab y 1 200 2 8.14 a 200 1 b 2 8.14 Step 3 Calculate the amount after 72 days. y 1 200 2 72 8. 14 0.43mg
Graph of y = e
Eample 4 Page 434, #18 Use the graph of y = e to evaluate each epression to four decimal places. Use your calculator too. 3 y e 20.0855
Continuously Compound Interest Formula Amount in account A Pe Principal Annual rate of interest rt Time in years
Eample 5 Page 434, #24 Find the amount in a continuously compounded account for the given conditions. principal: $2000 annual interest: 5.1% time: 3 yr A Pe rt 0.051(3) 2000e $2330.65
Eample A Suppose you invested $1050 at an annual rate of 5.5% compound continuously. How much you will have in the account after 5 years. Step 1 Find A, P, r and t A P r t? 1050 0.055 5
Eample A Step 2 Find A A Pe rt 1050e $1382.36 (0.055)(5) A P r t? 1050 0.055 5
Lesson 8-3 Logarithmic Functions as Inverses
Logarithmic Function y 10 y log
Logarithm The logarithm to the base b of a positive number y is defined as follows: y b log y b
Eample 2 Page 442, #6 Write the equation in logarithmic form. 2 49 7 y b log b y log7 49 2
Eample 2 Page 442, #12 Write the equation in logarithmic form. 3 1 1 3 27 y b log y b 1 3 1 log 3 27
Eample 3 Page 442, #14 Evaluate the logarithm log2 16 log216 log b y y b 16 2 4 2 2 4
Eample 3 Page 442, #15 Evaluate the logarithm log4 2 log4 2 log b y y b 2 4 2 2 2 2 2 2 1 2 1 2 2 2 1 2 1 2
Common Logarithm A common logarithm is a logarithm that use base 10. log10 y log y
Eample Page 442, #46 Use your calculator to evaluate the logarithm to four decimals places. Then find the largest integer that is less than the value of the logarithm. log17.52 1.2435 Largest integer is 1 10 1.2435 17.52
Eample Page 442, #58 Write the equation in eponential form. 1 log 2 3 9 log y y b b 1 3 9 2
Lesson 8-4 Properties of Logarithms
Properties of Logarithms Product Property log MN log M log N b b b Quotient Property M log log M log N b b b N Power Property log b M log M b
Eample 1 Page 449, #2, 4, and 8 State the property or properties used to write each epression. #2) log3 32 log3 8 log3 4 Quotient Property #4) log n p p 6 log6 n Power Property #8) 2 4 2logw 4logz logw z Power Property Product Property
Eample 2, Page 449, #14 Write each logarithmic epression as a single logarithm log8 2log6 log3 log8 log6 log3 log8 log36 log3 8 log log3 36 2 log log3 9 2
Eample 2, Page 449, #14 2 log 9 2 log 3 9 2 log 3 log3
Eample 2, Page 449, #18 Write each logarithmic epression as a single logarithm log log y log z 7 7 7 log y log 7 7 z log y 7 z
Eample 3 Page 449, #22 Epand the logarithm 4 2 log3m n 4 3 2 log m n log3m 4 2 logn 4 2 log3 logm logn log3 4logm 2logn
Eample 3 Page 449, #28 Epand the logarithm log 8 3a 8 5 log 8 log 3a 8 8 5 log 8 8 8 5 1 2 2 log 3 a log 8 log 3 log a 5 1 2 2 8 8 8
Eample 3 Page 449, #28 log 8 log 3 log a 5 1 2 2 8 8 8 1 5 log 8 log 3 log 2 2 8 8 8 a 1 5 1 log 3 log 2 2 8 8 a
Eample Page 449, #34 Use properties of logarithms to evaluate each epression. 3log 2 log 4 2 2 3() 1 log 2 2 2 1 32log 2 2 32( 1)
Eample Page 449, #38 Use the properties of logarithms to evaluate the epression. 1 2log84 log88 3 2 8 8 log 4 log 8 log 8 log8 2 16 2 16 3 1 3 log 8 1 8 1 3
Lesson 8-5, Part 1 Eponential and Logarithmic Equations
Eponential Equation Eponential Function Eponential Equation y b b c a
Eample 1 Page 456, #2 Solve the equation. Round your answer 4 decimal places. log419 4 19 log19 log4 2.1240
Eample 1 Page 456, #10 Solve the equation. Round your answer 4 decimal places. 2 1 25 144 log 144 2 1 25 log144 log25 2 1 log144 1 2 log25 log144 1 log25 2 2 2 0.5440 2 0.2720
Change of Base To evaluate a logarithm with any base, you can use the Change of Base Formula log b M logm logb
Eample 2 Page 456, #16 Use the Change of base formula to evaluate the epression. Then convert it to a logarithm in base 8 log 7 log7 2 log2 log 7 log 2 8 2. 8074 log 8 2.8074
Eample 2 Page 456, #16 2.8074 8 343.0046 log 7 is approimately equal to 2.8074 or log8 343 2
Eample 3 Page 456, #20 Use the Change of Base Formula to solve the equation 6 2 21 log6 21 2 log21 2 log6 log21 log6 2 1.6992 2 2 2 0.8496
Eample 3 Page 456, #26 Use the Change of Base Formula to solve the equation 2 4 89 2 log 4 log 89 4 4 ( 2) log 4 log 89 4 4 4 2 (1) log 89 2 log 89 4 2 log89 log4 log89 log4 5.2379 2
Lesson 8-5, Part 2 Eponential and Logarithmic Equations
Eample 6 Page 456, #34 Solve the equation. Check your answers. 2log 1 1 10 2 2log 1 2 2 1 log 2 0.3162
Eample 6 Page 456, #40 Solve the equation. Check your answers. 2log( 1) 5 2log( 1) 5 2 2 5 log( 1) 2 5 10 2 1 5 10 2 1 315.2
Eample 7 Page 457, #42 Solve the equation. log log3 8 log 8 3 8 10 3 8 10 3 310 8
Eample 7 Page 457, #46 Solve the equation. 3log log6 log2.4 9 3 log log6 log2.4 9 3 log log2. 4 9 6 3 2. 4 log 9 6
Eample 7 Page 457, #46 3 2. 4 log 9 6 log0.4 3 9 10 9 0.4 3 2.510 9 3 10 0.4 0.4 0.4 9 3 3 9 3 2.5 10 3 1357.2
Lesson 8-6 Natural Logs
Natural Logarithmic Function If y e, then log, which is commonly written as ln y. e y e 2.71828 y e ln y
Properties of Natural Logarithms Product Property lnmn lnm lnn Quotient Property M ln lnm ln N N Power Property lnm lnm
Eample 1 Page 464, #4 Write the epression as a single natural logarithm. 4ln8 ln10 4 ln8 ln10 4 ln8 (10) ln40960
Eample 1 Page 464, #8 Write the epression as a single natural logarithm. 1 1 ln ln y 4ln z ln 4ln 3 3 y z ln y 1 3 lnz 4 1 3 ln y z 4
Eample 1 Page 464, #8 1 3 ln y z 4 ln 3 z y 4
Eample 2 Page 464, #10 Find the value of y for the given value of y 15 3ln, for 7.2 y y 15 3ln 7.2 20.92
Eample 3 Page 465, #14 ln y Solve each equation ln3 6 3 e 6 6 3 e 3 3 e 6 3 134.48 y e
Eample 3 Page 465, #20 ln y Solve each equation y e 1 ln 4 2 1 4 e 2 4 1 e 2 e 4 21 110.20
Eample 4 Page 465, #24 Use natural logarithms to solve each equation. e 2 10 ln10 2 ln y y e ln10 2 2 2 1.1513
Eample 4 Page 465, #28 Use natural logarithms to solve each equation. 9 e 86 e 9 68 9(ln14) 23.752 ln y y e e 9 ln14 14 9
Eample Page 466, #62 Solve each equation ln( 2) ln 4 3 3 2e 4 2 ln 3 4 2 e 3 e 3 4 2 78.34 4