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4.3 - 1Copyright © 2013, 2009, 2005 Pearson Education, Inc. 1
4Inverse, Exponential, and Logarithmic Functions
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 2
4.3 Logarithmic Functions
• Logarithms• Logarithmic Equations• Logarithmic Functions• Properties of Logarithms
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 3
Logarithms
The previous section dealt with exponential functions of the form y = ax for all positive values of a, where a ≠ 1. The horizontal line test shows that exponential functions are one-to-one, and thus have inverse functions.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 4
Logarithms
The equation defining the inverse of a function is found by interchanging x and y in the equation that defines the function. Starting with y = ax and interchanging x and y yields
.yx a
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 5
Logarithms
yx a
Here y is the exponent to which a must be raised in order to obtain x. We call this exponent a logarithm, symbolized by the abbreviation “log.” The expression logax represents the logarithm in this discussion. The number a is called the base of the logarithm, and x is called the argument of the expression. It is read “logarithm with base a of x,” or “logarithm of x with base a.”
4.3 - 6Copyright © 2013, 2009, 2005 Pearson Education, Inc. 6
Logarithm
For all real numbers y and all positive numbers a and x, where a ≠ 1,
if and only if
The expression logax represents the exponent to which the base a must be raised in order to obtain x.
logay x .yx a
4.3 - 7Copyright © 2013, 2009, 2005 Pearson Education, Inc. 7
Example 1 WRITING EQUIVALENT LOGARITHMIC AND EXPONENTIAL FORMS
The table shows several pairs of equivalent statements, written in both logarithmic and exponential forms.
4.3 - 8Copyright © 2013, 2009, 2005 Pearson Education, Inc. 8
Logarithmic form: y = loga x
Exponent
Base
Exponential form: ay = x
Exponent
Base
Example 1 WRITING EQUIVALENT LOGARITHMIC AND EXPONENTIAL FORMS
To remember the relationships among a, x, and y in the two equivalent forms y = logax and x = ay, refer to these diagrams.
4.3 - 9Copyright © 2013, 2009, 2005 Pearson Education, Inc. 9
Example 2 SOLVING LOGARITHMIC EQUATIONS
Solve each equation.
Solution
(a)8
log 327x
8log 3
27x
Write in exponential form.
3 827
x
33 2
3x
38 227 3
23
x Take cube roots
4.3 - 10Copyright © 2013, 2009, 2005 Pearson Education, Inc. 10
Example 2 SOLVING LOGARITHMIC EQUATIONS
Check8
log 327x
Let x = 2/3.
Write in exponential form.
Original equation
2
?
3
8log 3
27
3 ?2 83 27
8 827 27
True
The solution set is 2.
3
4.3 - 11Copyright © 2013, 2009, 2005 Pearson Education, Inc. 11
Example 2 SOLVING LOGARITHMIC EQUATIONS
Solve each equation.
Solution
(b)
4
5log
2x
Write in exponential form.
254 x2 51(4 ) x ( )mn m na a
52 x
4
5log
2x
1 2 2 1 24 (2 ) 2
32 xThe solution set is {32}.
Apply the exponent.
4.3 - 12Copyright © 2013, 2009, 2005 Pearson Education, Inc. 12
Example 2 SOLVING LOGARITHMIC EQUATIONS
Solve each equation.
Solution
(c)
Write in exponential form.349 7x
1 32(7 ) 7x 2 1 37 7x
349log 7 x
12
3x
The solution set is
Write with the same base.
Power rule for exponents.
Set exponents equal.
16
x Divide by 2.
1.
6
4.3 - 13Copyright © 2013, 2009, 2005 Pearson Education, Inc. 13
Logarithmic Function
If a > 0, a ≠ 1, and x > 0, then
defines the logarithmic function with base a.
( ) logax xf
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Logarithmic Functions
Exponential and logarithmic functions are inverses of each other. The graph of y = 2x is shown in red. The graph of its inverse is found by reflecting the graph of y = 2x across the line y = x.
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 15
Logarithmic Functions
The graph of the inverse function, defined by y = log2 x, shown in blue, has the y-axis as a vertical asymptote.
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Logarithmic Functions
Since the domain of an exponential function is the set of all real numbers, the range of a logarithmic function also will be the set of all real numbers. In thesame way, both the range of an exponential function and the domain of a logarithmic function are the set of all positive real numbers. Thus, logarithms can be found for positive numbers only.
4.3 - 17Copyright © 2013, 2009, 2005 Pearson Education, Inc. 17
Domain: (0, ) Range: (– , )
LOGARITHMIC FUNCTION ( ) logax xf
x (x)
¼ – 2½ – 11 0
2 1
4 2
8 3 (x) = loga x, for a > 1, is
increasing and continuous on its entire domain, (0, ) .
For (x) = log2 x:
4.3 - 18Copyright © 2013, 2009, 2005 Pearson Education, Inc. 18
Domain: (0, ) Range: (– , )
LOGARITHMIC FUNCTION ( ) logax xf
x (x)
¼ – 2½ – 11 0
2 1
4 2
8 3The y-axis is a vertical asymptote
as x 0 from the right.
For (x) = log2 x:
4.3 - 19Copyright © 2013, 2009, 2005 Pearson Education, Inc. 19
Domain: (0, ) Range: (– , )
LOGARITHMIC FUNCTION ( ) logax xf
x (x)
¼ – 2½ – 11 0
2 1
4 2
8 3 The graph passes through the points
For (x) = log2 x:
1, 1 , 1,0 , and ,1 .a
a
4.3 - 20Copyright © 2013, 2009, 2005 Pearson Education, Inc. 20
Domain: (0, ) Range: (– , )
LOGARITHMIC FUNCTION ( ) logax xf
x (x)
¼ 2½ 11 0
2 – 14 – 28 – 3
(x) = loga x, for 0 < a < 1, is decreasing and continuous on its entire domain, (0, ) .
For (x) = log1/2 x:
4.3 - 21Copyright © 2013, 2009, 2005 Pearson Education, Inc. 21
Domain: (0, ) Range: (– , )
LOGARITHMIC FUNCTION ( ) logax xf
x (x)
¼ 2½ 11 0
2 – 14 – 28 – 3
For (x) = log1/2 x:
The y-axis is a vertical asymptote as x 0 from the right.
4.3 - 22Copyright © 2013, 2009, 2005 Pearson Education, Inc. 22
Domain: (0, ) Range: (– , )
LOGARITHMIC FUNCTION ( ) logax xf
x (x)
¼ 2½ 11 0
2 – 14 – 28 – 3
For (x) = log1/2 x:
The graph passes through the points 1
, 1 , 1,0 , and ,1 .aa
4.3 - 23Copyright © 2013, 2009, 2005 Pearson Education, Inc. 23
Characteristics of the Graph of ( ) logax xf
1. The points are on the graph.
2. If a > 1, then is an increasing function. If 0 < a < 1, then is a decreasing function.
3. The y-axis is a vertical asymptote.4. The domain is (0,), and the range is (– , ).
1, 1 , 1,0 , and ,1a
a
4.3 - 24Copyright © 2013, 2009, 2005 Pearson Education, Inc. 24
Example 3 GRAPHING LOGARITHMIC FUNCTIONS
Graph each function.
Solution
(a) 1 2( ) logx xf
First graph y = (½)x which defines the inverse function of , by plotting points. The graph of (x) = log1/2x is the reflection of the graph ofy = (½)x across the line y = x. The ordered pairs for y = log1/2x are found by interchanging the x- and y-values in the ordered pairs for y = (½)x .
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Example 3 GRAPHING LOGARITHMIC FUNCTIONS
Solution
1 2( ) logx xfGraph each function.
(a)
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Example 3 GRAPHING LOGARITHMIC FUNCTIONS
Solution
(b) 3( ) logx xf
Another way to graph a logarithmic function is to write (x) = y = log3 x in exponential form as x = 3y, and then select y-values and calculate corresponding x-values.
Graph each function.
4.3 - 27Copyright © 2013, 2009, 2005 Pearson Education, Inc. 27
Example 3 GRAPHING LOGARITHMIC FUNCTIONS
Solution
Graph each function.
(b) 3( ) logx xf
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Caution If you write a logarithmic function in exponential form to graph, as in Example 3(b), start first with y-values to calculate corresponding x-values. Be careful to write the values in the ordered pairs in the correct order.
4.3 - 29Copyright © 2013, 2009, 2005 Pearson Education, Inc. 29
Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and range.
Solution
(a) 2( ) log ( 1)x x f
The graph of (x) = log2(x – 1) is the graph of (x) = log2 x translated 1 unit to the right. The vertical asymptote has equation x = 1. Since logarithms can be found only for positive numbers, we solve x – 1 > 0 to find the domain, (1,) . To determine ordered pairs to plot, use the equivalent exponential form of the equation y = log2 (x – 1).
4.3 - 30Copyright © 2013, 2009, 2005 Pearson Education, Inc. 30
Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and range.
Solution
2( ) log ( 1)x x f
2log ( 1)y x
1 2yx Write in exponential form.
2 1yx Add 1.
(a)
4.3 - 31Copyright © 2013, 2009, 2005 Pearson Education, Inc. 31
Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and range.
Solution
2( ) log ( 1)x x f
We first choose values for y and then calculate each of the corresponding x-values. The range is (– , ).
(a)
4.3 - 32Copyright © 2013, 2009, 2005 Pearson Education, Inc. 32
Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and range.
Solution
(b) 3( ) (log ) 1x x f
The function (x) = (log3 x) – 1 has the same graph as g(x) = log3 x translated 1 unit down. We find ordered pairs to plot by writing the equation y = (log3 x) – 1 in exponential form.
4.3 - 33Copyright © 2013, 2009, 2005 Pearson Education, Inc. 33
Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and range.
Solution
3( ) log ( 1)x x f
3(log ) 1y x
31 logy x Add 1.
13yx Write in exponential form.
(b)
4.3 - 34Copyright © 2013, 2009, 2005 Pearson Education, Inc. 34
Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and range.
Solution
3( ) log ( 1)x x f
Again, choose y-values and calculate the corresponding x-values. The domain is (0, ) and the range is (– , ).
(b)
4.3 - 35Copyright © 2013, 2009, 2005 Pearson Education, Inc. 35
Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and range.
Solution
(c) 4( ) log ( 2) 1x x f
The graph of f(x) = log4(x + 2) + 1 is obtained by shifting the graph of y = log4x to the left 2 units and up 1 unit. The domain is found by solving x + 2 > 0, which yields (–2, ). The vertical asymptote has been shifted to the left 2 units as well, and it has equation x = –2. The range is unaffected by the vertical shift and remains (– , ).
4.3 - 36Copyright © 2013, 2009, 2005 Pearson Education, Inc. 36
Example 4 GRAPHING TRANSLATED LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and range.
Solution
(c) 4( ) log ( 2) 1x x f
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 37
Properties of Logarithms
The properties of logarithms enable us to change the form of logarithmic statements so that products can be converted to sums, quotients can be converted to differences, and powers can be converted to products.
4.3 - 38Copyright © 2013, 2009, 2005 Pearson Education, Inc. 38
Properties of Logarithms
For x > 0, y > 0, a > 0, a ≠ 1, and any real number r, the following properties hold.
DescriptionThe logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.
Property
Product Property
log log loga a axy x y
4.3 - 39Copyright © 2013, 2009, 2005 Pearson Education, Inc. 39
Properties of Logarithms
For x > 0, y > 0, a > 0, a ≠ 1, and any real number r, the following properties hold.
Quotient Property
log log loga a a
xx y
y
DescriptionThe logarithm of the quotient of two numbers is equal to the difference between the logarithms of the numbers.
Property
4.3 - 40Copyright © 2013, 2009, 2005 Pearson Education, Inc. 40
Properties of Logarithms
Power Property
log logra ax r x
DescriptionThe logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the number.
For x > 0, y > 0, a > 0, a ≠ 1, and any real number r, the following properties hold.
Property
4.3 - 41Copyright © 2013, 2009, 2005 Pearson Education, Inc. 41
Properties of Logarithms
Logarithm of 1log 1 0a
DescriptionThe base a logarithm of 1 is 0.
The base a logarithm of a is 1.
For x > 0, y > 0, a > 0, a ≠ 1, and any real number r, the following properties hold.
Property
Base a Logarithm of alog 1aa
4.3 - 42Copyright © 2013, 2009, 2005 Pearson Education, Inc. 42
Example 5 USING THE PROPERTIES OF LOGARITHMS
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
(a) 6log (7 9)
6 6 6log ( ) l7 7og9 9log Product property
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Example 5 USING THE PROPERTIES OF LOGARITHMS
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
(b) 9
15log
7
9 9 9log log15
15 77
log Quotient property
4.3 - 44Copyright © 2013, 2009, 2005 Pearson Education, Inc. 44
Example 5 USING THE PROPERTIES OF LOGARITHMS
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
(c) 5log 8
51 2
5 5log 8 log (8 ) l12
og 8 Power property
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Example 5 USING THE PROPERTIES OF LOGARITHMS
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
(d) 2 4loga
mnqp t
2 42 4log log log log log log( )a a a a a a
mnqm n q p t
p t
log log log (2 log 4 log )a a a a am n q p t log log log 2 log 4 loga a a a am n q p t
Use parentheses to avoid errors.
Be careful with signs.
4.3 - 46Copyright © 2013, 2009, 2005 Pearson Education, Inc. 46
Example 5 USING THE PROPERTIES OF LOGARITHMS
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
(e) 3 2loga m
3 2 2 3log l2
og log3a a am m m
4.3 - 47Copyright © 2013, 2009, 2005 Pearson Education, Inc. 47
Example 5 USING THE PROPERTIES OF LOGARITHMS
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Power property
3 51logb m
x yn z
3 51log log log m
b b bx y zn
Product and quotient properties
13 5 3 5
log logn
nb bm m
x y x yz z
1 nn a a
3 5
log nb m
x yz
(f)
Solution
4.3 - 48Copyright © 2013, 2009, 2005 Pearson Education, Inc. 48
Example 5 USING THE PROPERTIES OF LOGARITHMS
Rewrite each expression. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
(f)3 5
log nb m
x yz
Power property 13 log 5 log logb b bx y m z
n
Distributive property3 5
log log logb b b
mx y z
n n n
4.3 - 49Copyright © 2013, 2009, 2005 Pearson Education, Inc. 49
Example 6 USING THE PROPERTIES OF LOGARITHMS
Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
(a) 3 3 3log ( 2) log log 2x x
3 3 3 3log log log log2)
( 2)2
( 2x xx x
Product and quotient properties
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Example 6 USING THE PROPERTIES OF LOGARITHMS
Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
(b) 2 log 3 loga am n
32log log l3 og o2 l ga a a an m nm Power property
2
3loga
mn
Quotient property
4.3 - 51Copyright © 2013, 2009, 2005 Pearson Education, Inc. 51
Example 6 USING THE PROPERTIES OF LOGARITHMS
Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
(c) 21 3log log 2 log
2 2b b bm n m n
21 3log log 2 log
2 2b b bm n m n
Power property1 22 3 2( )log log 2 logb b bm n m n
4.3 - 52Copyright © 2013, 2009, 2005 Pearson Education, Inc. 52
Example 6 USING THE PROPERTIES OF LOGARITHMS
Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
21 3log log 2 log
2 2b b bm n m n
Product and quotient properties
1 2 3 2
2
(2 )logb
m nm n
3 2 1 2
3 2
2logb
nm
Rules for exponents
(c)
4.3 - 53Copyright © 2013, 2009, 2005 Pearson Education, Inc. 53
Example 6 USING THE PROPERTIES OF LOGARITHMS
Write each expression as a single logarithm with coefficient 1. Assume all variables represent positive real numbers, with a ≠ 1 and b ≠ 1.
Solution
21 3log log 2 log
2 2b b bm n m n
1 23
3
2logb
nm
Rules for exponents
3
8logb
nm
Definition of a1/n
(c)
4.3 - 54Copyright © 2013, 2009, 2005 Pearson Education, Inc. 54
Caution There is no property of logarithms to rewrite a logarithm of a sum or difference. That is why, in Example 6(a), log3(x + 2) was not written as log3 x + log3 2. The distributive property does not apply in a situation like this because log3 (x + y) is one term. The abbreviation “log” is a function name, not a factor.
4.3 - 55Copyright © 2013, 2009, 2005 Pearson Education, Inc. 55
Example 7USING THE PROPERTIES OF LOGARITHMS WITH NUMERICAL VALUES
Assume that log10 2 = 0.3010. Find each logarithm.
Solution
(a) 10log 4
102 log 2
102
10log log4 2
2(0.3010)
0.6020
4.3 - 56Copyright © 2013, 2009, 2005 Pearson Education, Inc. 56
Example 7USING THE PROPERTIES OF LOGARITHMS WITH NUMERICAL VALUES
Assume that log10 2 = 0.3010. Find each logarithm.
Solution
(b) 10log 5
10 10log log10
52
0.6990
0. 01 301
110 0log 10 log 2
4.3 - 57Copyright © 2013, 2009, 2005 Pearson Education, Inc. 57
Theorem on Inverses
For a > 0, a ≠ 1, the following properties hold.
log (for 0) and loga x xaa x x a x
Copyright © 2013, 2009, 2005 Pearson Education, Inc. 58
Theorem on Inverses
The following are examples of applications ofthis theorem.
The second statement in the theorem will be useful in Sections 4.5 and 4.6 when we solve other logarithmic and exponential equations.
7 10log7 ,10 535log ,3 and 1log 1r
kr k