Transcript Logarithmic Functions

10TH
EDITION
COLLEGE
ALGEBRA
LIAL
HORNSBY
SCHNEIDER
4.3 - 1
4.3
Logarithmic Functions
Logarithms
Logarithmic Equations
Logarithmic Functions
Properties of Logarithms
4.3 - 2
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.
4.3 - 3
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
x  ay .
4.3 - 4
Logarithms
x a
y
Here y is the exponent to which a must be raised
in order to obtain x. We call this exponent a
logarithm, symbolized by “log.” The
expression loga x 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 - 5
Logarithm
For all real numbers y and all positive numbers a
and x, where a ≠ 1,
y  loga x
if and only if
x  ay .
A logarithm is an exponent. The expression
loga x represents the exponent to which the
base “a” must be raised in order to obtain x.
4.3 - 6
Logarithms
Exponent
Logarithmic form: y = loga x
Base
Exponent
Exponential form: ay = x
Base
4.3 - 7
Logarithms
4.3 - 8
Example 1
SOLVING LOGARITHMIC
EQUATIONS
Solve
8
a. logx
3
27
8
Solution logx
3
27
8
3
x 
27
2
3
x  
3
2
x
3
Write in exponential
form.
3
8 2
 
27  3 
3
Take cube roots
4.3 - 9
Example 1
Check:
SOLVING LOGARITHMIC
EQUATIONS
8
logx
3
27
8
log2 3
3
27
Original equation
? Let x = ⅔.
3
8
2
  
 3  27
8
8

27 27
The solution set is
? Write in
exponential form
True

2
.
3
4.3 - 10
Example 1
SOLVING LOGARITHMIC
EQUATIONS
Solve
5
b. log4 x 
2
Solution
5
log4 x 
2
45 2  x
(4 )  x
12 5
The solution
set is {32}.
25  x
Write in exponential
form.
a mn  (a m )n
41 2  (22 )1 2  2
32  x
4.3 - 11
SOLVING LOGARITHMIC
EQUATIONS
Example 1
Solve
c. log49 3 7  x
Solution
49  7
x
3
(7 )  7
2 x
The solution
set is 1 .

6
13
72 x  71 3
1
2x 
3
1
x
6
Write in exponential form.
Write with the same base.
Power rule for exponents.
Set exponents equal.
Divide by 2.
4.3 - 12
Logarithmic Function
If a > 0, a ≠ 1, and x > 0, then
f ( x )  loga x
defines the logarithmic function with
base a.
4.3 - 13
Logarithmic Function
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 across the
line y = x.
4.3 - 14
Logarithmic Function
The graph of the
inverse function,
defined by y = log2 x,
shown in blue, has the
y-axis as a vertical
asymptote.
4.3 - 15
Logarithmic Function
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 the
same way, both the range of an
exponential function and the domain of
a logarithmic function are the set of all
positive real numbers, so logarithms
can be found for positive numbers
only.
4.3 - 16
LOGARITHMIC FUNCTION f ( x )  loga x
Domain: (0, )
Range: (– , )
For (x) = log2 x:
x
(x)
¼
–2
½
–1
1
2
4
8
0
1
2
3
 (x) = loga x, a > 1, is
increasing and continuous on
its entire domain, (0, ) .
4.3 - 17
LOGARITHMIC FUNCTION f ( x )  loga x
Domain: (0, )
Range: (– , )
For (x) = log2 x:
x
(x)
¼
–2
½
–1
1
2
4
8
0
1
2
3
 The y-axis is a vertical asymptote
as x  0 from the right.
4.3 - 18
LOGARITHMIC FUNCTION f ( x )  loga x
Domain: (0, )
Range: (– , )
For (x) = log2 x:
x
(x)
¼
–2
½
–1
1
2
4
8
0
1
2
3
 The graph passes through the points
1 
 , 1 , 1,0  , and  a,1 .
a

4.3 - 19
LOGARITHMIC FUNCTION f ( x )  loga x
Domain: (0, )
Range: (– , )
For (x) = log1/2 x:
x
(x)
¼
2
½
1
1
2
4
8
0
–1
–2
–3
 (x) = loga x, 0 < a < 1, is
decreasing and continuous on its
entire domain, (0, ) .
4.3 - 20
LOGARITHMIC FUNCTION f ( x )  loga x
Domain: (0, )
Range: (– , )
For (x) = log1/2 x:
x
(x)
¼
2
½
1
1
2
4
8
0
–1
–2
–3
 The y-axis is a vertical
asymptote as x  0 from the
right.
4.3 - 21
LOGARITHMIC FUNCTION f ( x )  loga x
Domain: (0, )
Range: (– , )
For (x) = log1/2 x:
x
(x)
¼
2
½
1
1
2
4
8
0
–1
–2
–3
 The graph passes through
the points  1 , 1 , 1,0  , and  a,1 .

a


4.3 - 22
Characteristics of the Graph
of f ( x )  loga x
1 
 , 1 , 1,0  , and  a,1
a

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
(–, ).
4.3 - 23
Example 2
GRAPHING LOGARITHMIC
FUNCTIONS
Graph the function.
a. f ( x )  log1 2 x
Solution
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
y = (½)x across the line y = x. The ordered pairs
for are found by interchanging the x- and yvalues in the ordered pairs for y = (½)x .
4.3 - 24
Example 2
GRAPHING LOGARITHMIC
FUNCTIONS
Graph the function.
a. f ( x )  log1 2 x
Solution
4.3 - 25
Example 2
GRAPHING LOGARITHMIC
FUNCTIONS
Graph the function.
f ( x )  log3 x
b.
Solution
Another way to graph a logarithmic function is
to write (x) = y = log3 x in exponential form as
x = 3 y.
4.3 - 26
Example 2
GRAPHING LOGARITHMIC
FUNCTIONS
Graph the function.
f ( x )  log3 x
a.
Solution
4.3 - 27
Caution If you write a logarithmic
function in exponential form, choosing yvalues to calculate x-values, be careful to
write the values in the ordered pairs in
the correct order.
4.3 - 28
Example 3
GRAPHING TRANSLATED
LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and
range.
a. f ( x )  log2 ( x  1)
Solution
The graph of (x) = log2 (x – 1) is the graph of
(x) = log2 x translated 1 unit to the right. The vertical
asymptote is x = 1. The domain of this function is
(1, ) since logarithms can be found only for positive
numbers. To find some ordered pairs to plot, use the
equivalent exponential form of the equation
y = log2 (x – 1).
4.3 - 29
GRAPHING TRANSLATED
LOGARITHMIC FUNCTIONS
Example 3
Graph each function. Give the domain and
range.
a. f ( x )  log2 ( x  1)
Solution
y  log2 ( x  1)
x 1 2
y
x  2 1
y
Write in exponential
form.
Add 1.
4.3 - 30
Example 3
GRAPHING TRANSLATED
LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and
range.
a. f ( x )  log2 ( x  1)
Solution
We choose values for y
and then calculate each
of the corresponding xvalues. The range is
(–, ).
4.3 - 31
Example 3
GRAPHING TRANSLATED
LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and
range.
b. f ( x )  (log3 x )  1
Solution
The function defined by (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 y = (log3 x) – 1
in exponential form.
4.3 - 32
Example 3
GRAPHING TRANSLATED
LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and
range.
b. f ( x )  (log3 x )  1
Solution
y  (log3 x )  1
y  1  log3 x
x 3
y 1
Add 1.
Write in exponential form.
4.3 - 33
Example 3
GRAPHING TRANSLATED
LOGARITHMIC FUNCTIONS
Graph each function. Give the domain and
range.
b. f ( x )  (log3 x )  1
Solution
Again, choose yvalues and calculate
the corresponding xvalues. The domain is
(0, ) and the range is
(–, ).
4.3 - 34
Properties of Logarithms
Since a logarithmic statement can be
written as an exponential statement, it is
not surprising that the properties of
logarithms are based on the properties of
exponents. The properties of logarithms
allow 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 - 35
Properties of Logarithms
For x > 0, y > 0, a > 0, a ≠ 1, and any real
number r:
Property
Product Property
loga xy  loga x  loga y
Description
The logarithm of the
product of two numbers is
equal to the sum of the
logarithms of the numbers.
4.3 - 36
Properties of Logarithms
For x > 0, y > 0, a > 0, a ≠ 1, and any real
number r:
Property
Quotient Property
x
loga  loga x  loga y
y
Description
The logarithm of the
quotient of two numbers
is equal to the difference
between the logarithms of
the numbers.
4.3 - 37
Properties of Logarithms
For x > 0, y > 0, a > 0, a ≠ 1, and any real
number r:
Property
Power Property
loga x  nloga x
n
Description
The logarithm of a
number raised to a power
is equal to the exponent
multiplied by the
logarithm of the number.
4.3 - 38
Properties of Logarithms
Two additional properties of logarithms
follow directly from the definition of loga x
since a0 = 1 and a1 = a.
loga1  0 and logaa  1
4.3 - 39
USING THE PROPERTIES OF
LOGARITHMS
Rewrite each expression. Assume all variables
represent positive real numbers, with a ≠ 1 and b ≠ 1.
Example 4
a. log6 (7 9)
Solution
log6 (7 9)  log6 7  log6 9
Product property
4.3 - 40
USING THE PROPERTIES OF
LOGARITHMS
Rewrite each expression. Assume all variables
represent positive real numbers, with a ≠ 1 and b ≠ 1.
Example 4
15
b. log9
7
Solution
15
log9
 log915  log9 7
7
Quotient property
4.3 - 41
USING THE PROPERTIES OF
LOGARITHMS
Rewrite each expression. Assume all variables
represent positive real numbers, with a ≠ 1 and b ≠ 1.
Example 4
c. log5 8
Solution
log5
1
8  log5 (8 )  log5 8
2
12
Power property
4.3 - 42
USING THE PROPERTIES OF
LOGARITHMS
Rewrite each expression. Assume all variables
represent positive real numbers, with a ≠ 1 and b ≠ 1.
Example 4
mnq
d. loga 2 4
Use parentheses
pt
to avoid errors.
Solution
mnq
2
4
loga 2 4  loga m  loga n  logaq  (loga p  loga t )
pt
 logam  logan  logaq  (2 loga p  4 loga t )
 logam  logan  logaq  2 loga p  4 loga t
Be careful with
signs.
4.3 - 43
USING THE PROPERTIES OF
LOGARITHMS
Rewrite each expression. Assume all variables
represent positive real numbers, with a ≠ 1 and b ≠ 1.
Example 4
3
e. loga m
2
Solution
loga m  loga m
3
2
23
2
 loga m
3
Power property
4.3 - 44
USING THE PROPERTIES OF
LOGARITHMS
Rewrite each expression. Assume all variables
represent positive real numbers, with a ≠ 1 and b ≠ 1.
Example 4
f.
logb
n
1n
x y 
x y
 logb  m 
m
z
 z 
3
5
3
5
3
1
x y
 logb m
n
z
n
a  a1 n
5
1
  logb x 3  logb y 5  logb z m 
n
Power property
Product and quotient
properties
4.3 - 45
USING THE PROPERTIES OF
LOGARITHMS
Rewrite each expression. Assume all variables
represent positive real numbers, with a ≠ 1 and b ≠ 1.
Example 4
f.
logb
n
1n
x y 
x y
 logb  m 
m
z
 z 
3
5
3
5
Solution
1
  3 logb x  5 logb y  m logb z 
n
3
5
m
 logb x  logb y  logb z
n
n
n
Power property
Distributive property
4.3 - 46
USING THE PROPERTIES OF
LOGARITHMS
Write the expression as a single logarithm with
coefficient 1. Assume all variables represent
positive real numbers, with a ≠ 1and b ≠ 1.
Example 5
a. log3 ( x  2)  log3 x  log3 2
Solution
( x  2)x
log3 ( x  2)  log3 x  log3 2  log3
2
Product and quotient
properties
4.3 - 47
USING THE PROPERTIES OF
LOGARITHMS
Write the expression as a single logarithm with
coefficient 1. Assume all variables represent
positive real numbers, with a ≠ 1and b ≠ 1.
Example 5
b. 2 loga m  3 loga n
Solution
2 loga m  3 loga n  loga m2  loga n 3
Power property
2
m
 loga 3
n
Quotient property
4.3 - 48
USING THE PROPERTIES OF
LOGARITHMS
Write the expression as a single logarithm with
coefficient 1. Assume all variables represent
positive real numbers, with a ≠ 1and b ≠ 1.
Example 5
1
3
c.
logb m  logb 2n  logb m 2n
2
2
Solution
1
3
logb m  logb 2n  logb m 2n
2
2
 logb m
12
 logb (2n )
32
 logb m n
2
Power properties
4.3 - 49
USING THE PROPERTIES OF
LOGARITHMS
Write the expression as a single logarithm with
coefficient 1. Assume all variables represent
positive real numbers, with a ≠ 1and b ≠ 1.
Example 5
1
3
c.
logb m  logb 2n  logb m 2n
2
2
Solution
m1 2 (2n )3 2
Product and quotient
 logb
2
properties
mn
32
12
2 n
 logb
32
m
Rules for exponents
4.3 - 50
USING THE PROPERTIES OF
LOGARITHMS
Write the expression as a single logarithm with
coefficient 1. Assume all variables represent
positive real numbers, with a ≠ 1and b ≠ 1.
Example 5
1
3
c.
logb m  logb 2n  logb m 2n
2
2
Solution
12
3
2 n
Rules for exponents
 logb  3 
m 
 logb
8n
3
m
Definition of a1/n
4.3 - 51
Caution There is no property of
logarithms to rewrite a logarithm of a sum or
difference. That is why, in Example 5(a),
log3(x + 2) was not written as log3 x + log3 2.
Remember, log3 x + log3 2 = log3(x • 2).
The distributive property does not apply in a
situation like this because log3 (x + y) is one term;
“log” is a function name, not a factor.
4.3 - 52
Example 6
USING THE PROPERTIES OF
LOGARITHMS WITH NUMERICAL
VALUES
Assume that log10 2 = .3010. Find each logarithm.
a. log10 4
Solution
log10 4  log10 2  2 log10 2  2(.3010)  .6020
2
b. log10 5
10
log10 5  log10
 log10 10  log10 2  1  .3010  .6990
2
4.3 - 53
Theorem on Inverses
For a > 0, a ≠ 1:
a
loga x
 x and logaa  x.
x
4.3 - 54
Theorem on Inverses
By the results of this theorem,
log7 10
7
 10, log5 5  3, and
3
logr r
k 1
 k  1.
The second statement in the theorem will be
useful in Sections 4.5 and 4.6 when we solve
other logarithmic and exponential equations.
4.3 - 55