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Exponential and Logarithmic Functions
Chapter 9
Angel, Intermediate Algebra, 7ed 2
9.1 – Composite and Inverse Functions
9.2 – Exponential Functions
9.3 – Logarithmic Functions
9.4 – Properties of Logarithms
9.5 – Common Logarithms
9.6 – Exponential and Logarithmic Equations
9.7 – Natural Exponential and Natural Logarithmic Functions
Chapter Sections
§ 9.1
Composite and Inverse Functions
Angel, Intermediate Algebra, 7ed 4
Composite Functions
The composite function is defined as
.( )( ) [ ( )] f g x f g xf g
Example:
Given f(x) = x2 – 3, and g(x) = x + 2, find .( )( )f g x
2( )( ) ( ) ( ) 3f g x f g x g x
2( 2) 3x 2 4 1x x x + 2 is substituted into each x in f(x).
2( 4 4) 3x x
g(x) is substituted into each x in f(x).
Angel, Intermediate Algebra, 7ed 5
Composite Functions
Example:
Given f(x) = x2 – 3, and g(x) = x + 2, find .( )( )g f x
( )( ) ( ) ( ) 2g f x g f x f x
2 3 2x 2 1x
x2 - 3 is substituted into each x in g(x).
f(x) is substituted into each x in g(x).
Angel, Intermediate Algebra, 7ed 6
One-to-One Functions
For a function to be one-to-one, it must not only pass the vertical line test, but also the horizontal line test.
A function is a one-to-one function if each value in the range corresponds with exactly one value in the domain.
x
y
Function
x
y
Not a one-to-one function
x
y
One-to one function
Angel, Intermediate Algebra, 7ed 7
Inverse Functions
Function: {(2, 6), (5,4), (0, 12), (4, 1)}
If f(x) is a one-to-one function with ordered pairs of the form (x,y), its inverse function, f -1(x), is a one-to-one function with ordered pairs of the form (y,x).
Inverse Function: {(6, 2), (4,5), (12, 0), (1, 4)}
• Only one-to-one functions have inverse functions.
• Note that the domain of the function becomes the range of the inverse function, and the range becomes the domain of the inverse function.
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Inverse Functions
1. Replace f(x) with y.
2. Interchange the two variables x and y.
3. Solve the equation for y.
4. Replace y with f –1(x). (This gives the inverse function using inverse function notation.)
To Find the Inverse Function of a One-to-One Function
Example:
Find the inverse function of
Graph f(x) and f(x) –1 on the same axes.
.1, 1f x x x
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Inverse Functions
1, 1f x x
1y x
1x y
22 1x y
2 1x y
2 1x y
1 2( ) 1, 0f x x x
Replace f(x) with y.
Interchange x and y.
Solve for y.
Replace y with f –1(x) .
Example continued:
Angel, Intermediate Algebra, 7ed 10
Inverse Functions
1, 1f x x x
1 2( ) 1, 0f x x x
Note that the symmetry is about the line y = x.
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1 2( )( ) 1 1f f x x
1f x x
Composites and Inverses
If two functions f(x) and f –1(x) are inverses of each other, . 1 1( )( ) and ( )( )f f x x f f x x
Example:
Show that .
1, 1f x x x 1 2( ) 1, 0f x x x and
.1 1( )( ) and ( )( ) f f x x f f x x
2x x
1 2( ) 1f x x
21( ) 1 1f x x
1 1x x
§ 9.2
Exponential Functions
Angel, Intermediate Algebra, 7ed 13
Exponential Functions
For any real number a > 0 and a 1,
f(x) = ax
is an exponential function.
For all exponential functions of this form,
1. The domain of the function is
2. The range of the function is
3. The graph passes through the points
.( , ) .(0, )
.1( 1, ), 0,1 , 1,aa
Angel, Intermediate Algebra, 7ed 14
Exponential Graphs
Example:
Graph the function f(x) = 3x.
1-1,3
0,1
1,3
Domain: Range: {y|y > 0}
Angel, Intermediate Algebra, 7ed 15
Exponential Graphs
Domain: Range: {y|y > 0} -1, 3
0,1 11,3
Notice that each graph passes through the point (0, 1).
Example:
Graph the function f(x) = 1 .3
x
§ 9.3
Logarithmic Functions
Angel, Intermediate Algebra, 7ed 17
Exponential Functions
For all positive numbers a, where a 1,
y = logax means x = ay.
y = logax
logarithm(exponent)
base
number
means x = ay
number base
exponent
Angel, Intermediate Algebra, 7ed 18
Exponential Functions
Exponential Form Logarithmic Form
50 = 1 log101= 0
23 = 8 log28= 3
41 1=2 16 1 2
1log = 416
-2 16 =36 6
1log = -236
Angel, Intermediate Algebra, 7ed 19
Logarithmic Functions
For all logarithmic functions of the form y = logax or f(x) = logax, where a > 0, a 1, and x > 0,
1. The domain of the function is .
2. The range of the function is .
3. The graph passes through the points
( , )
(0, )
1( , 1), 1,0 , ,1 .aa
Angel, Intermediate Algebra, 7ed 20
Logarithmic Graphs
Range: Domain: {x|x > 0}
Graph the function f(x) = log10x.
1 , -110
1,0
10,1
Notice that the graph passes through the point (1,0).
Example:
Angel, Intermediate Algebra, 7ed 21
Exponential vs. Logarithmic Graphs
Exponential Function Logarithmic Functiony = ax (a > 0, a 1)
y = logax (a > 0, a 1)
, 0,
11,a
, 0,
1 , 1a
1,a
1,0
,1a
Domain:
Range:
Points on Graph: x becomes y
y becomes x
0,1
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f(x) = log10x
f(x) = 10x
Notice that the two graphs are inverse functions.
f(x)
f -1(x)
Exponential vs. Logarithmic Graphs
§ 9.4
Properties of Logarithms
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Product Rule
For positive real numbers x, y, and a, a 1,Product Rule for Logarithms
Example:
log5(4 · 7) = log54 + log57
log10(100 · 1000) = log10100 + log101000 = 2 + 3 = 5
log log loga a axy x y
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Quotient Rule
For positive real numbers x, y, and a, a 1,Quotient Rule for Logarithms
Example:
log log loga a ax x yy
7 7 710log log 10 log 22
10 10 101log log 1 log 1000 0 3 3
1000
Property 1
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Power Rule
If x and y are positive real numbers, a 1, and n is any real number, then
Power Rule for Logarithms
Example:
log logna ax n x
49 9log 3 4log 3
210 10log 100 2log 100 2 2 4
Property 2
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Additional Properties
If a > 0, and a 1, Additional Properties of Logarithms
Example:
log xa a x
49log 9 4
610log 10 6
Property 5
Property 4log ( 0)xaa x x
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Combination of Properties
Example:
6 6 65log ( 3) [2 log ( 4) 3log ]x x x
Write the following as the logarithm of a single expression.
5 2 36 6 6log ( 3) [log ( 4) log ]x x x Power Rule
5 2 36 6log ( 3) [log ( 4) ]x x x Product Rule
5
6 2 3( 3)log
( 4)x
x x
Quotient Rule
§ 9.5
Common Logarithms
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Common Logarithms
The common logarithm of a positive real number is the exponent to which the base 10 is raised to obtain the number.
If log N = L, then 10L = N.
The antilogarithm is the same thing as the inverse logarithm.
If log N = L, then N = antilog L.
log 962 = 2.98318
Number Exponent
antilog 2.98318 = 962
NumberExponent
Example:
§ 9.6
Exponential and Logarithmic Equations
Angel, Intermediate Algebra, 7ed 32
Properties
a. If x = y, ax = ay.
b. If ax = ay, then x = y.
c. If x = y, then logbx = logby (x > 0, y > 0).
d. If logbx = logby, then x = y (x > 0, y > 0).
Properties for Solving Exponential and Logarithmic Equations
Properties 6a-6d
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Solving Equations
Example:
Solve the equation 4 256.x
2 82 2x Rewrite each side with the same base.
2 82 2x
2 8x Property 6b.
4x Solve for x.
Angel, Intermediate Algebra, 7ed 34
Example:
Solve the equation log( 3) log log 4.x x Product Rule
Property 6d.
log( 3) log 4x x
( 3) 4x x 2 3 4x x
2 3 4 0x x
( 4)( 1) 0x x
4 or 1x x
Check:
Stop! Logs of negative numbers are not real numbers.
log( 3) lo4 4g( ) log 4.
log( 3) log( ) l1 1 og 4. log 4 0 log 4
log 4 log 4 True
Solving Equations
§ 9.7
Natural Exponential and Natural Logarithmic
Functions
Angel, Intermediate Algebra, 7ed 36
Definitions
The natural exponent function is
f(x) = ex
where e 2.71823.
Natural logarithms are logarithms to the base e. Natural logarithms are indicated by the letters ln.
logex = ln x
Example:
ln 1 = 0 (e0 = 1) ln e = 1 (e1 = e)
Angel, Intermediate Algebra, 7ed 37
Change of Base Formula
For any logarithm bases a and b, and positive number x,Change of Base Formula
loglog
logb
ab
xx
a
This is very useful because common logs or natural log can be found using a calculator.
Example:
3log 198 log198log3
2.29670.4771
4.8136 Note that the natural log could have also been used.ln198
ln 3 5.2883
1.0986 4.8136
Angel, Intermediate Algebra, 7ed 38
Properties
Notice that these are the same properties as those for the common logarithms.
Properties for Natural Logarithmsln ln ln ( 0 and 0)
ln ln ln ( 0 and 0)
ln ln ( 0)n
xy x y x y
x x y x yy
x n x x
Product Rule
Power Rule
Quotient Rule
Additional Properties for Natural Logarithms and Natural Exponential Expressions
Property 7
Property 8ln
ln =x
( 0)
x
x
e
e x x
Angel, Intermediate Algebra, 7ed 39
Solving Equations
Example: Solve the following equation.
ln( 3) ln( 3) ln 40x x
ln[( 3)( 3)] ln 40x x Product Rule
2ln( 9) ln 40x Simplify
2 9 40x Property 6d
2 49x Solve for x.
7x Check solutions in original equation. (You will notice that only the positive 7 yields a true statement.)
Angel, Intermediate Algebra, 7ed 40
Applications
In 2000, a lake had 300 trout. The growth in the number of trout is estimated by the function g(t) = 300e0.07t where t is the number of years after 2000. How many trout will be in the lake in a) 2003? b) 2010?
In the year 2000, t = 0. (Notice that f(0) =300e0.07(0) = 300e0 = 300, the original number of trout.)
In the year 2003, t = 3. g(3) = 300e0.07(3) = 300e0.21 = 300(1.2337) 370 trout in 2003.
In the year 2010, t = 10. g(10) = 300e0.07(10) = 300e0.70 = 300(2.0138) 604 trout in 2010.
Example: