Chapter 8wellsmat.startlogic.com/sitebuildercontent/sitebuilderfiles/alg2... · Exploring...
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Chapter 8 Exponential and Logarithmic Functions
Transcript of Chapter 8wellsmat.startlogic.com/sitebuildercontent/sitebuilderfiles/alg2... · Exploring...
Chapter 8
Exponential and Logarithmic Functions
Lesson 8-1
Exploring Exponential Models
Exponential Function
The general form of an exponential function is y = abx.
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
Example 4 – Page 427, #16, 20
Without graphing determine whether each function representsexponential growth or exponential decay.
129(1.63)xy
Exponential Growth
54
6
x
y
Exponential Decay
Example 1 – Page 426, #4
Graph each Function.
9(3)xy
0 9
1 3
2 1
6 0.0123
x y
Exponential Growth
Example 5 – Page 427, #26
Graph each Function.
0.25xy
1 4
0.5 0.5
1 0.25
x y
Exponential Decay
Asymptote
An asymptote is a line that a graph approaches as x or yincreases in absolute value.
Example 2 – A
Carl’s weight at 12 yr is 82 lb. Assume that his weightincreases at a rate of 16% each year. Write an exponentialfunction to model the increase. Calculate his weight after5 yr.
Step 1 – Find a and b.
1b r 0.161
1.16
xy ab
82a
a is the original
b is the growth factoror decay factor
Example 2 – A
Step 2 – Find the exponential function.
xy ab821.16
ab
82(1.16)xy
Step 3 – Calculate his weight after 5 yr.
582(1.16)y
172.228
If the model is correct, Carlwill weight about 172 lb in 5 yr.
Example 2 – B
A motorcycle purchased for $9000 today it will be worth 6%less each year. For what can you expect to sell the motorcycle at the end of 5 yr?
Step 1 – Find a and b.
1b r 0.061
0.94
xy ab
9000a
a is the original
b is the growth factoror decay factor
Example 2 – B
Step 2 – Find the exponential function.
xy ab90000.94
ab
9000(0.94)xy
Step 3 – Calculate the sale price after 5 yr.
59000(0.94)y
$6605.14
Lesson 8-2
Properties of Exponential Functions
Example 1 – Page 434, #2
Graph each function. Label the asymptote of each graph
1
2
x
y
0 1
1 0.50
2 0.25
3 0.125
x y
asymptote is y = 0
Example 3 – Page 434, #16
Iodine-131 is used to find leaks in water pipes. It has ahalf-life of 8.14 days. Write the exponential decay functionfor a 200-mg sample. Find the amount of iodine-131 remaining after 72 days.
Step 1 – Find a and b.
8.141
2
x
b
xy ab
200a
a is the original
b is ½ and x is the number of days
Example 3 – Page 434, #16
8.141
2
x
b
200a Step 2 – Find the exponential function.
xy ab
8.141200
2
x
y
Step 3 – Calculate the amount after 72 days.
.2
8 147
1200
2y
0.43mg
Graph of y = ex
Example 4 – Page 434, #18
Use the graph of y = ex to evaluate each expression tofour decimal places. Use your calculator too.
3y e
20.0855
Continuously Compound Interest Formula
rtA Pe
Amount inaccount
Principal
Annual rate of interest
Time in years
Example 5 – Page 434, #24
Find the amount in a continuously compounded accountfor the given conditions.
principal: $2000annual interest: 5.1%time: 3 yr
rtA Pe0.051(3)2000
$2330.65e
Example – 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
?10500.0555
APrt
Example – A
Step 2 – Find A?10500.0555
APrt
rtA Pe
(0.055)(5)1050e
$1382.36
Lesson 8-3
Logarithmic Functions
as Inverses
Logarithmic Function
10xy logy x
Logarithm
The logarithm to the base b of a positive number y is
defined as follows:
xy b logb y x
Example 2 – Page 442, #6
Write the equation in logarithmic form.
249 7
xy b logb y x
7log 49 2
Example 2 – Page 442, #12
Write the equation in logarithmic form.
31 1
3 27
xy b logb y x
1
3
1log 3
27
Example 3 – Page 442, #14
Evaluate the logarithm 2log 16
2log 16 x
logb y x xy b
16 2x
42 2x
4 x
Example 3 – Page 442, #15
Evaluate the logarithm 4log 2
4log 2 x
logb y x xy b
2 4x
22 2x
1 2x
22 2 x
1 2
2 2
1
2
x
x
1
2x
Common Logarithm
A common logarithm is a logarithm that use base 10.
10log y logy
Example – 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
1.243510 17.52
Example – Page 442, #58
Write the equation in exponential form.
3
1log 2
9
xy blogb y x
213
9
Lesson 8-4
Properties of Logarithms
Properties of Logarithms
log log logb b b
MM N
N
Product Property log log logb b bMN M N
Quotient Property
Power Property log logb bxM x M
Example 1 – Page 449, #2, 4, and 8
State the property or properties used to write each expression.
#2)3 3 3log 32 log 8 log 4 Quotient Property
#4)
#8)
6 6log logn p p
x xn
Power Property
Power Property
Product Property
2 42log 4log logw z w z
Example 2, Page 449, #14
Write each logarithmic expression as a single logarithm
log8 2log6 log3
2log8 log log3
log8 log36
6
log3
log log3
2log log3
9
8
36
Example 2, Page 449, #14
2
log log39
2log 3
9
2log
3
Example 2, Page 449, #18
Write each logarithmic expression as a single logarithm
7 7 7log log logx y z 7 7log logxy z
7logxy
z
Example 3 – Page 449, #22
4 2log3m n4
2
3log
m
n
Expand the logarithm
4 2log3 log logm n
log3 4log 2logm n
4 2log3 logm n
Example 3 – Page 449, #28
58log 8 3a
58 8log 8 log 3a
Expand the logarithm
512 2
8 8log 8 log 3 a
512 2
8 8 8log 8 log 3 log a
Example 3 – Page 449, #28
512 2
8 8 8log 8 log 3 log a
8 8 8
1 5log 8 log 3 log
2 2a
8 8
1 51 log 3 log
2 2a
Example – Page 449, #34
Use properties of logarithms to evaluate each expression.
2 23log 2 log 4 2
23 l( ) og 21
23 2log 2
(3 2
1
1)
Example – Page 449, #38
Use the properties of logarithms to evaluate the expression.
8 8
12log 4 log 8
3
132
8 8log 4 log 8
13
83
8
16log
2
16log
2
8log 8 1
Lesson 8-5, Part 1
Exponential and Logarithmic Equations
Exponential Equation
cxb axy b
Exponential Function Exponential Equation
Example 1 – Page 456, #2
Solve the equation. Round your answer 4 decimal places.
4 19x
4log 19 x
log19
log4x
2.1240x
Example 1 – Page 456, #10
Solve the equation. Round your answer 4 decimal places.
2 125 144x
25log 144 2 1x
log144
2 1log25
x
log144
2log
125
x
log1441
2log2
2
5
2
x
0.5440
2x
0.2720x
Change of Base
To evaluate a logarithm with any base, you can use
the Change of Base Formula
loglog
logb
b
MM
Example 2 – Page 456, #16
Use the Change of base formula to evaluate the expression.
Then convert it to a logarithm in base 8
2log 7 log7
log2
2 8log 7 log x
82 l. 074 og8 x
2.8074
Example 2 – Page 456, #16
2.80748 x
343.0046x
is approximately equal to 2.8074 or 2log 7 8log 343
Example 3 – Page 456, #20
Use the Change of Base Formula to solve the equation
26 21x
6log 21 2x
log21
2log6
x
log21
2log
2
6
2
x
1.6992
2
0.8496
x
x
Example 3 – Page 456, #26
Use the Change of Base Formula to solve the equation
24 89x 2
4 4log l 9og4 8x
4 4log 4( 2) log 89x
4
4
2 (1) log 89
2 log 89
x
x
log89
2log4
x
log89
log42x
5.2379x
Lesson 8-5, Part 2
Exponential and Logarithmic Equations
Example 6 – Page 456, #34
Solve the equation. Check your answers.
2log 1x
2 2
2log 1
1log
2
x
x
1210 x
0.3162x
Example 6 – Page 456, #40
Solve the equation. Check your answers.
2log( 1) 5x
2 2
2log( 1) 5
5log( 1)
2
x
x
5210 1x
315.2x
5
210 1 x
Example 7 – Page 457, #42
Solve the equation.
log log3 8x
log 83
x
8103
x
810 3 x
83 10x
Example 7 – Page 457, #46
3log log6 log2.4 9x
3log log6 log2.4 9x 3
log log2 4 96
.x
Solve the equation.
3 2.log 9
4
6
x
Example 7 – Page 457, #46
3 2.log 9
4
6
x
3log0.4 9x
9 310 0.4x
9 3
9 3
0.4 0
10 0.4
2.5 10
.4
x
x
33 9 32.5 10 x
1357.2x
Lesson 8-6
Natural Logs
Natural Logarithmic Function
xy e
If then which is commonly written as,xy e log ,e y x ln .y x
lny x
2.71828e
Properties of Natural Logarithms
ln ln lnM
M NN
Product Property ln ln lnMN M N
Quotient Property
Power Property ln lnxM x M
Example 1 – Page 464, #4
Write the expression as a single natural logarithm.
4ln8 ln10 4ln8 ln10
4ln8 (10)
ln40960
Example 1 – Page 464, #8
Write the expression as a single natural logarithm.
1
ln ln 4ln3
x y z 1
ln 4ln3
xy z
13 4ln lnxy z
13
4ln
xy
z
Example 1 – Page 464, #8
13
4ln
xy
z
3
4ln
xy
z
Example 2 – Page 464, #10
Find the value of y for the given value of x
15 3ln ,y x for 7.2x
7.215 3ln
20.92
y
y
Example 3 – Page 465, #14
Solve each equation
ln3 6x
ln
x
x
e
y
y
63x e
63
3 3
x e
6
3
134.48
ex
x
Example 3 – Page 465, #20
Solve each equation
1ln 4
2
x
ln
x
x
e
y
y
41
2
xe
41 2x e
4 2 1
110.20
x e
x
Example 4 – Page 465, #24
Use natural logarithms to solve each equation.
2 10xe
ln
x
x
e
y
y
ln10 2x
ln10 2
2 2
x
1.1513x
Example 4 – Page 465, #28
Use natural logarithms to solve each equation.
9 8 6x
e
ln
x
x
e
y
y
9
9
6 8
14
x
x
e
e
ln149
x
9(ln14)
23.752
x
x
Example – Page 466, #62
Solve each equation
ln( 2) ln 4 3x
2ln 3
4
x
32
4
xe
32 4x e
3 4 2
78.34
x e
x
