Slide 12- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Exponential and Logarithmic Functions
CHAPTER
12.1 Composite and Inverse Functions12.2 Exponential Functions12.3 Logarithmic Functions12.4 Properties of Logarithms12.5 Common and Natural Logarithms12.6 Exponential and Logarithmic
Equations with Applications
1122
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Composite and Inverse Functions12.112.1
1. Find the compososition of two functions.2. Show that two functions are inverses.3. Show that a function is one-to-one.4. Find the inverse of a function. 5. Graph a given function’s inverse function.
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Composite functions: If f and g are functions, then the composition of f and g is defined asFor all x in the domain g for which g(x) is the domain of f. The composition of g and f is defined as
for all x in the domain of g for which f(x) is in the domain of g.
f g x f g x
g f x g f x
Slide 12- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
If and find the following.
Solution
Example
( ) 3 8f x x ( ) 2 5,g x x
f g x
f g x f g x
2 5f x
53 2 8x
6 15 8x
6 7x
Replace g(x) with 2x – 5.
In f(x), replace x with 2x – 5.
Simplify.
a. b. g f x
a.
Slide 12- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
g f x g f x
3 8g x
6 16 5x
6 11x
Replace f(x) with 3x + 8.
In g(x), replace x with 3x + 8
Simplify.
b.
82 3 5x
Slide 12- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Inverse functions: Two functions f and g are inverses if and only if for all x in the domain of g and for all x in the domain of f.
f g x x g f x x
Inverse FunctionsTo determine whether two functions f and g are inverses of each other,
1. Show that for all x in the domain of g. 2. Show that for all x in the domain of f.
f g x x g f x x
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Solution
Example
Verify that f and g are inverses.
3( ) 4 3, ( )
4
xf x x g x
We need to show that and f g x x .g f x x 3
4
xf
3
44 3
x
3 3x
f g x
x
g f x 4 3g x
4
4 3 3x
4
4
x
x
Since and , f and g are inverses. f g x x g f x x
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One-to-One function: A function f is one-to-one if for any two numbers a and b in its domain, when f(a) = f(b), a = b and when a b, f(a) f(b).
Horizontal Line Test for One-to-One FunctionsGiven a function’s graph, the function is one-to-one if every horizontal line that can intersect the graph does so at one and only one point.
Slide 12- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Determine whether the graph is a one-to-one function.
Solution
Example
2 5f x x
X
Y
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-2
2
4
6
8
10
12
14
0X
Y
-5 -4 -3 -2 -1 1 2 3 4 5
-4
-2
2
4
6
8
10
12
14
0
A horizontal line can intersect the graph in more than one point, so the function is not one-to-one.
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Finding the Inverse Function of a One-to-One Function
1. If necessary, replace f(x) with y. 2. Replace all x’s with y’s and y’s with x’s.3. Solve the equation from step 2 for y.4. Replace y with f-1(x).
Existence of Inverse FunctionsA function has an inverse function if and only if the function is one-to-one.
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Solution
Example
Find f -1(x) for the function f(x) = 7x – 4.
y = 7x – 4
x = 7y – 44
7
xy
1 4
7
xf x
Since f[f -1(x)]= x and f -1[f (x)] = x, they are inverses.
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Graph of Inverse FunctionsThe graphs of f and f-1 are symmetric to the graph of y = x.
Slide 12- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solution
Example
Sketch the inverse of the function whose graph is shown.
Draw the line y = x and reflect the graph on the line.
X
Y
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
0
X
Y
-10 -8 -6 -4 -2 2 4 6 8 10
-10
-8
-6
-4
-2
2
4
6
8
10
0
Slide 12- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
If f(x) = x + 7 and g(x) = 2x – 12, what is
a) -44
b) -3
c) 3
d) 44
4 .f g
Slide 12- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
If f(x) = x + 7 and g(x) = 2x – 12, what is
a) -44
b) -3
c) 3
d) 44
4 .f g
Slide 12- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find f -1(x) for 6x – 7.
a)
b)
c)
d)
7
6
x
7
6
x
6
7
x
6
7
x
Slide 12- 18 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Find f -1(x) for 6x – 7.
a)
b)
c)
d)
7
6
x
7
6
x
6
7
x
6
7
x
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Exponential Functions12.212.2
1. Define and graph exponential functions.2. Solve equations of the form bx = by for x.3. Use exponential functions to solve application problems.
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Exponential function: If b > 0, b 1, and x is any real number, then the exponential function is f(x) = bx.
Note: The definition of the exponential function has two restrictions on b. If b = 1, then f(x) = bx = 1x = 1, which is a linear function. If b < 0, then we could get values for which the function is not defined as a real number.
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ExampleGraph f(x) = 2x, and g(x) = 3x.. Solution
x –2 –1 0 1 2
f(x) 1 2 4
g(x) 1 3 9
1
41
2
Comparing the graphs, we can see the greater value of b, the steeper the graph.
1
9
1
3
f(x) = 2xg(x) = 3x
X
Y
-10-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10
-10-9-8-7-6-5-4-3-2-1
123456789
10
0
Slide 12- 22 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
ExampleGraph f(x) = 3x+3 .
Solutionx f(x)
-4 3-1 =
-3 30 = 1
-2 31 = 3
-1 32 = 9
0 33 = 27
1
3
f(x) = 3x+3
X
Y
-24 -21 -18 -15 -12 -9 -6 -3 3
3
6
9
12
15
18
21
24
27
0
Slide 12- 23 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Exponential Equations1. If necessary, write both sides of the equation as a
power of the same base.2. If necessary, simplify the exponents.3. Set the exponents equal to each other.4. Solve the resulting equation.
The One-to-One Property of ExponentialsGiven b > 0 and b 1, if bx = by, then x = y.
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Example
Solve.
a. 3x = 81 b. 2 14
16x
Solution
a. 3x = 81
3x = 34
x = 4
The solution set is {4}.
b. 2 14
16x
2 24 4x
2 2x
4x
The solution set is {–4}.
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Example
Rayanne deposited $15,000 in an account that pays 7% annual interest compounded quarterly. How much is accumulated in the account after 10 years?
SolutionUnderstand We are asked to find A and given that
t = 10, P = $15,000, r = 0.07, and n = 4.
Plan Use the formula 1 .nt
rA P
n
Slide 12- 26 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
1
tn
A Pr
n
4(10)0.07
5 11 004
,0A
4015,000 1 0.0175A
$30,023.96A
Substitution.
Simplify.
Evaluate using a calculator and round to the nearest cent.
Execute
Answer After 10 years, the accumulated account is $30,023.96.
Slide 12- 27 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
4(10)0.07
30,023.96 14
P 40
30,023.96 1.0175P
14,999.99993 P
Check Verify that the principal is $15,000 if the accumulated amount is $30,023.96 after the principal is compounded quarterly.
40
30,023.96
1.0175P
Since the accumulated amount was rounded, it is expected our calculated value of the principal to be slightly different from $15,000.
Slide 12- 28 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Which of the following functions does the graph represent?
a) f(x) = 4x+2
b) f(x) = 4x–2
c) f(x) = 2x+4
d) f(x) = 2x–4
Slide 12- 29 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Which of the following functions does the graph represent?
a) f(x) = 4x+2
b) f(x) = 4x–2
c) f(x) = 2x+4
d) f(x) = 2x–4
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A stereo system is purchased for $2500. It’s value each year is about 85% of its value in the preceding year. Its value in dollars after t years is given by the exponential function V(t) = 2500(0.85)t. Find the salvage value of the stereo after 4 yr.
a) $1285.23
b) $1296.42
c) $1305.02
d) $1316.84
Slide 12- 31 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
A stereo system is purchased for $2500. It’s value each year is about 85% of its value in the preceding year. Its value in dollars after t years is given by the exponential function V(t) = 2500(0.85)t. Find the salvage value of the stereo after 4 yr.
a) $1285.23
b) $1296.42
c) $1305.02
d) $1316.84
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Logarithmic Functions12.312.3
1. Convert between exponential and logarithmic forms.2. Solve logarithmic equations by changing to
exponential form.3. Graph logarithmic functions.4. Solve applications involving logarithms.
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Logarithm: If b > 0 and b 1, then y = logbx is equivalent to x = by.
x = by y = logbx
The exponent is the logarithm.
The base is the base of the logarithm.
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Solving Logarithmic EquationsTo solve an equation of the form logbx = y, where b, x, or y is a variable, write the equation in exponential form, by = x, and then solve for the variable.
Example
Solve log7 x = 3.
SolutionWrite the equation in exponential form. log7 x = 3
73 = x 343 = x
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ExampleSolve
Solution Begin by rewriting the equation in exponential form and then solve.
Divide both sides by -3 to isolate the logarithm.
Simplify.
Subtract 11 from both sides.
211 3log 4x
211 3log 4x
23log 15x
2log 5x 52x
32x
Write in exponential form.
For any real number b, where b > 0 and b 1, 1. logbb = 1 2. log b 1 = 0
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Graphing Logarithmic FunctionsTo graph a function in the form f(x) = logbx,
1. Replace f(x) with y and then write the logarithm in exponential form x = by.
2. Find ordered pairs that satisfy the equation by assigning values to y and finding x.
3. Plot the ordered pairs and draw a smooth curve through the points.
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ExampleGraph f(x) = log1/4x.
SolutionReplacing f(x) by y, we see that the equation y = log1/4 x can be rewritten in exponential form as
y
–2
–1
0
1
2
14 .
yx
14
yx
214 16
114 4
014 1
11 14 4
21 14 16
x = 0
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ExampleThe function P = 95 – 30 log2x models the percent, P, of students who recall the important features of a classroom lecture over time, where x is the number of days that have elapsed since the lecture was given. What percent of the students recall the important features of a lecture 8 days after it was given? (Source: Psychology for the New Millennium, 8th Edition, Spencer A. Rathos, Thomson Publishing Company)
SolutionUnderstand We are given the function that models the
percent, P, of students who recall the important features of a lecture x days after it is given. We are to find the percent of students who recall the important features of a lecture 8 days after it was given.
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continued
Execute
Answer 5% of the students remember the important features of a lecture 8 days after it is given.
Plan Evaluate P = 95 – 30log2 x where x = 8.
P = 95 – 30log2 8
P = 95 – 30(3)
P = 95 – 90
P = 5
Check 5 = 95 – 30log2 x
–90 = – 30log2 x
3 = log2 x
23 = x
8 = x
Slide 12- 40 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Write 62 = 36 in logarithmic form.
a)
b)
c)
d)
6log 2 36
6log 36 2
2log 6 36
2log 36 6
Slide 12- 41 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Write 62 = 36 in logarithmic form.
a)
b)
c)
d)
6log 2 36
6log 36 2
2log 6 36
2log 36 6
Slide 12- 42 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Write in exponential form.
a)
b)
c)
d)
3
1log 4
81
4 13
81
3 14
81
4 13
81
381 4
Slide 12- 43 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Write in exponential form.
a)
b)
c)
d)
3
1log 4
81
4 13
81
3 14
81
4 13
81
381 4
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Solve log3x = 5.
a) -243
b) -125
c) 125
d) 243
Slide 12- 45 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve log3x = 5.
a) -243
b) -125
c) 125
d) 243
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Properties of Logarithms12.412.4
1. Apply the inverse properties of logarithms.2. Apply the product, quotient, and power properties of
logarithms.
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Inverse Properties of LogarithmsFor any real numbers b and x, where b > 0 and b 1,
and x > 0,1. blog
bx = x 2. logbbx= x
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Further Properties of LogarithmsFor real numbers x, y, and b, where x > 0, y > 0, b > 0,
and b 1.Product Rule of Logarithms: logbxy = logbx + logby
Quotient Rule of Logarithms:
Power Rule of Logarithms: logbxr = rlogbx
(The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.)
log log logb b b
xx y
y
(The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.)
(The logarithm of a number raised to the power is equal to the exponent times the logarithm of the number.)
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Example
Use the product rule of logarithms to write each expression as a sum of logarithms.
a. logbpqr b. logb x(x + 5)
Solution
a. logbpqr = logbp + logbq + logbr
b. logb x(x+5) = logbx + logb(x + 5)
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Example
Use the quotient rule of logarithms in the form to write the expression as a
single logarithm.
Solution
log log logb b b
xx y
y
6 6log 108 log x
6 6log 108 log x 6
108log
x
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Example
Use the power rule of logarithms to write the expression as a multiple of a logarithm.
Solution
a. log3x11
log3x11 = 11 log3x
b. 7
1logb x
a.
b. 7
1logb x
= logbx-7 = -7logbx
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Example
Write the expression as a sum or difference of multiples of logarithms.
Solution
Use the product and quotient rules.
= logb x7+ logb z9 logb y4
7 9
4logb
x z
y
Use the power rule.
7 9
4logb
x z
y
= 7 logb x + 9 logb z 4 logb y
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Example
Write the expression as a single logarithm. Leave answer in simplest form without negative or fractional exponents.
Solution log 5 log 4a ax x
log 5 log 4a ax x log 5 4a x x
2log 20a x x
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Use the power rule to write the expression as a multiple of a logarithm.
a)
b)
c)
d)
log 8a
1log
8 a
1log 2
8 a
1log 8
2 a
1log
16 a
Slide 12- 55 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use the power rule to write the expression as a multiple of a logarithm.
a)
b)
c)
d)
log 8a
1log
8 a
1log 2
8 a
1log 8
2 a
1log
16 a
Slide 12- 56 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Express as a single logarithm:
a)
b)
c)
d)
log 2log 5log .a a ax y z
2
5loga
xy
z
2 5loga
x
y z2
5loga
y
xz
log 2 5a x y z
Slide 12- 57 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Express as a single logarithm:
a)
b)
c)
d)
log 2log 5log .a a ax y z
2
5loga
xy
z
2 5loga
x
y z2
5loga
y
xz
log 2 5a x y z
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Common and Natural Logarithms12.512.5
1. Define common logarithms and evaluate them using a calculator.
2. Solve applications using common logarithms.3. Define natural logarithms and evaluate them using a
calculator.4. Solve applications using natural logarithms.
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Common logarithms: Logarithms with a base of 10. Log10x is written as log x.
Solution
Example
Use a calculator to approximate each common logarithm. Round to the nearest thousandth if necessary. a. log 456 b. log 0.00257
a. log 456
b.
2.659
log 0.00257 2.590
Slide 12- 60 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
ExampleThe function is used to calculate sound
intensity, where d represents the intensity in decibels, I represents the intensity watts per unit of area, and I0 represents the faintest audible sound to the average human ear, which is 10-12 watts per square meter. What is the intensity level of sounds at a decibel level of 75 dB?
010log
Id
I
Understand We are given the function, and we are to find the sound intensity, I. 0
10log ,I
dI
Slide 12- 61 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
Execute
Answer The sound intensity is 10–4.5.
Plan Using , substitute 75 for d and
10-12 for I0 and then solve for I.0
10logI
dI
1275 10 log
10
I
127.5 log
10
I
7.5
1210
10
I
12 12 7.512
10 10 1010
I
4.510 I
Slide 12- 62 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
Check If the sound intensity is 10-4.5, verify the decibel reading is 75.
4.5
1210
10log10
d
7.510log10d
10 7.5d
75d
Slide 12- 63 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Natural logarithms: Base-e logarithms are called natural logarithms and logex is written as ln x. Note that ln e = 1.
Solution
ExampleUse a calculator to approximate each natural logarithm to four decimal places. a. ln 67 b. ln 0.0072
a. ln 67
b. ln 0.00257 5.9638
4.2047
Slide 12- 64 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
ExampleIf an account pays 8% annual interest, compounded continuously, how long will it take a deposit of $25,000 to produce an account balance of $100,000?Understand We are to find the time it takes for $25,000
to grow to $100,000 if it is compounded continuously at 8%.
1ln
At
r PPlan In , replace P with 25,000, r with
0.08, A with $100,000, and then simplify.
Slide 12- 65 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
17.33t
Answer The account balance will reach $100,000 in approximately 17.33 years.
Substitute.
Divide.
Approximate using a calculator.
Execute1 100,000
ln0.08 25,000
t
1ln 4
0.08t
Slide 12- 66 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
Check
117.33 ln
0.08 25,000
A
1.3864 ln25,000
A
1.3864 ln ln 25,000A 1.3864 ln 25,000 ln A
11.513 ln A11.513e A
100,007.5 A
Because 17.33 was not the exact time, $100,007.45 is reasonable.
Slide 12- 67 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use a calculator to approximate log 0.267 to four decimal places.
a) -1.3205
b) -0.5735
c) 1.3060
d) 1.8493
Slide 12- 68 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use a calculator to approximate log 0.267 to four decimal places.
a) -1.3205
b) -0.5735
c) 1.3060
d) 1.8493
Slide 12- 69 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use a calculator to approximate ln 21 to four decimal places.
a) 0.7636
b) 1.3316
c) 3.0445
d) 5.3688
Slide 12- 70 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use a calculator to approximate ln 21 to four decimal places.
a) 0.7636
b) 1.3316
c) 3.0445
d) 5.3688
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Exponential and Logarithmic Equations with Applications12.612.6
1. Solve equations that have variables as exponents.2. Solve equations containing logarithms.3. Solve applications involving exponential and
logarithmic functions. 4. Use the change-of-base formula.
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Properties for Solving Exponential and Logarithmic Equations
For any real numbers b, x, and y, where b > 0 and b 1,1. If bx = by, then x = y. 2. If x = y, then bx = by. 3. For x > 0 and y > 0, if logbx = logby, then x = y.
4. For x > 0 and y > 0, if x = y, then logbx = logby.
5. For x > 0, if logbx = y, then by = x.
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ExampleSolve 8x = 15.Solution
log8 log15x
log8 log15x
Use if x = y, then logbx = logby (property 4).
log15
log8x
1.3023x
Divide both sides by log 8.
The exact solution is . Using a calculator, we find .
log15
log8x
Check 8x = 15
81.3023 = 15
15.0001 = 15
The answer is correct.
Slide 12- 74 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solving Equations Containing LogarithmsTo solve equations containing logarithms, use the
properties of logarithms to simplify each side of the equation and then use one of the following.
If the simplification results in an equation in the form logbx = logby, use the fact that x = y, and then solve for the variable.
If the simplification results in an equation in the form logbx = y, write the equation in exponential form, by = x, and then solve for the variable (as we did in Section 10.3).
Slide 12- 75 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
Solve
Solution32 4x
8 4x
12 x
2log ( 4) 3.x
This equation is in the form logbx = y, so write it in exponential form, by = x.
Solve for x.
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ExampleSolveSolution
3 3 3log log ( 3) log 10.x x
Logarithms are defined for positive numbers only. A check will show that x = 5 the only solution.
3 3log ( 3) log 10x x ( 3) 10x x
2 3 10x x 2 3 10 0x x
5 2 0x x
5 0 or 2 0x x 5 or 2x x
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ExampleSolve.Solution
ln( 4) ln( 1) ln(8 )x x x
20 8 12x x 4ln ln(8 )
1
xx
x
4
81
xx
x
(8 )( 1) 4x x x
28 8 4x x x x
29 8 4x x x
0 ( 6)( 2)x x
6 2x or x
These 2 solutions check and are the solutions for this equation.
Slide 12- 78 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
If $1000 is deposited into an account at 7.6% interest compounded continuously, how much money will be in the account after 8 years?
Understand We are given P = $1000, r = 0.076, and t = 8 and we are asked to find A.
Plan Use .rtA Pe
Execute 0.076(8)1000A e0.6081000A e
$1836.75A
Slide 12- 79 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
There will be $1836.75 in the account. Answer
Check1
ln .A
tr P
Use the formula
1 1836.75ln .
0.076 1000t
8t It checks.
Slide 12- 80 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
ExampleA nuclear reactor contains 200 grams of radioactive plutonium 239P. Plutonium disintegrates according to the formula A = A0e-0.0000284t. How much will remain after 5000 years?
Understand We are given A0 = 200 g and t = 5000 years and are asked to find A. Plan Use
Execute
0.00002840 .tA A e
0.0000284 5000200A e
173.52A
0.142200A e Simplify.
Substitute.
Slide 12- 81 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
continued
About 173.52 grams will remain after 5000 years.
Answer
Check Use A = A0e-0.0000284t to verify that it takes 5000 years for 200 grams of 239P to disintegrate to 173.52 grams.
5000.9 tIt checks.
173.52 = 200e-0.0000284t
0.0000284173.52
200te
0.0000284173.52ln ln
200te
173.52ln 0.0000284
200t
173.52ln
2000.0000284
t
Slide 12- 82 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Change-of-Base FormulaIn general, if a > 0, a 1, b > 0, b 1, and x > 0, then
In terms of common and natural logarithms,
log
log .log
ba
b
xx
a
log lnlog .
log lna
x xx
a a
Slide 12- 83 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Example
Use the change-of-base formula to calculate log512. Round the answer to four decimal places.
Solution
5
log12log 12
log5
1.5440
Check1.54405 12.0009 12 The answer is correct.
Slide 12- 84 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve log x + log(x – 3) = 1.
a) x = -2
b) x = 5
c) x = -2 or x = 5
d) x is undefined.
Slide 12- 85 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Solve log x + log(x – 3) = 1.
a) x = -2
b) x = 5
c) x = -2 or x = 5
d) x is undefined.
Slide 12- 86 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
If $500 is deposited into an account at 9% interest compounded continuously, how much will be in the account after 5 years?
a) $653.79
b) $784.16
c) $892.36
d) $45,008.57
Slide 12- 87 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
If $500 is deposited into an account at 9% interest compounded continuously, how much will be in the account after 5 years?
a) $653.79
b) $784.16
c) $892.36
d) $45,008.57
Slide 12- 88 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use the change-of-base formula to approximate log756.
a) 0.48
b) 0.85
c) 2.07
d) 8
Slide 12- 89 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Use the change-of-base formula to approximate log756.
a) 0.48
b) 0.85
c) 2.07
d) 8
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