Section 4.3
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Transcript of Section 4.3
Section 4.3
Logarithmic Functions and Graphs
Flashback
Consider the graph of the exponential function y = f(x) = 3x.
• Is f(x) one-to-one?• Does f(x) have an inverse that is a function?• Find the inverse.
Inverse of y = 3x
f (x) = 3x
y = 3x
x = 3y
x = 3y
Now, solve for y.
y= the power to which 3 must be raised in order to obtain x.
x = 3y
Solve for y.
y= the power to which 3 must be raised in order to obtain x.
Symbolically, y = log 3 x
“The logarithm, base 3, of x.”
Logarithm
For all positive numbers a, where a 1,
Logax is an exponent to which the base a
must be raised to give x.
yaa =x isequivalent to y=logx
Logarithmic Form
Exponential Form
a exponenlog x y t
base
ya
exponent
a e
x
b s
All a log is . . . is an exponent!
Argument(always positive)
Logarithmic Functions• Logarithmic functions are inverses of
exponential functions.
Graph: x = 3y or y = log 3 x 1. Choose values for y.2. Compute values for x.3. Plot the points and connect them with a
smooth curve.
* Note that the curve does not touch or cross
the y-axis.
Logarithmic Functions continued
(1/27, 3)31/27
(1/9, 2)21/9
(1/3, 1)11/3
2
1
0
y
(9, 2)9
(3, 1)3
(1, 0)1
(x, y)x = 3y
Graph: x = 3y
y = log 3 x
Side-by-Side Comparison
f (x) = 3x f (x) = log 3 x
Comparing Exponential and Logarithmic Functions
Logarithmic Functions
• Remember: Logarithmic functions are inverses of exponential functions.
x
- 1a
The inverse of f(x) = a
is given by
f (x)=log x
Asymptotes
• Recall that the horizontal asymptote of the exponential function y = ax is the x-axis.
• The vertical asymptote of a logarithmic function is the y-axis.
ay=log x
Logarithms
• Convert each of the following to a logarithmic equation.a) 25 = 5x b) ew = 30
log
A logarithm is an exponent!
ya x y x a
Example
• Convert each of the following to an exponential equation.
• a) log7 343 = 3
log7 343 = 3 7 3 = 343
b) logb R = 12
The logarithm is the exponent.
The base remains the same.
Finding Certain Logarithms• Find each of the following.
a) log2 16 b) log10 1000
c) log16 4 d) log10 0.001
Common Logarithm
10
For allpositivenumbersx,
logx=log x
•Log button on your calculator
is the common log *
Logarithms, base 10, are called common logarithms.
Example• Find each of the following common
logarithms on a calculator. Round to four decimal places.
a) log 723,456b) log 0.0000245c) log (4)
Does not existERR: nonreal anslog (4)
4.61084.610833916log 0.0000245
5.85945.859412123log 723,456
RoundedReadoutFunction Value
Natural Logarithms•Logarithms, base e, are called
natural logarithms.
• The abbreviation “ln” is generally used for natural logarithms.
• Thus, ln x means loge x.
* ln button on your calculator
is the natural log *
Example• Find each of the following natural logarithms
on a calculator. Round to four decimal places.
a) ln 723,456b) ln 0.0000245c) ln (4)
Does not existERR: nonreal ansln (4)
10.616810.61683744ln 0.0000245
13.491813.49179501ln 723,456
RoundedReadoutFunction Value
Changing Logarithmic Bases
•The Change-of-Base FormulaFor any logarithmic bases a and b,
and any positive number M,
loglog .
loga
ba
MM
b
Use change of base formula when you have a logarithm that is not base 10 or e.
Example
Find log6 8 using common logarithms.
Solution: First, we let a = 10, b = 6, and M = 8. Then we substitute into the change-of-base formula:
10
16
0
loglog
l
8
og
1.1606
68
ExampleWe can also use base e for a conversion.
Find log6 8 using natural logarithms.
Solution: Substituting e for a, 6 for b and 8 for M, we have
6 6
loglog
log
ln81.1606
ln 6
88 e
e
Properties of Logarithms
a
a
For a>0, a 1
2.
For the logarithmic base e,
3.
loga=1
log 1=0
ln e=1
l 04. n1=
1.
Graphs of Logarithmic Functions
• Graph: y = f(x) = log6
x.– Select y.– Compute x.
21/36
11/6
3216
236
16
01
yx, or 6 y
Example• Graph each of the following. • Describe how each graph can be obtained
from the graph of y = ln x. • Give the domain and the vertical
asymptote of each function.
• a) f(x) = ln (x 2)
• b) f(x) = 2 ¼ ln x
• c) f(x) = |ln (x + 1)|
Graph f(x) = ln (x 2)• The graph is a shift 2 units
right.
• The domain is the set of all real numbers greater than 2.
• The line x = 2 is the vertical
asymptote.
1.0995
0.6934
03
0.6932.5
1.3862.25
f(x)x
Graph f(x) = 2 ¼ ln x• The graph is a vertical
shrinking, followed by a reflection across the x-axis, and then a translation up 2 units.
• The domain is the set of all positive real numbers.
• The y-axis is the vertical asymptote.
14
1.5985
1.7253
21
2.1730.5
2.5760.1
f(x)x
Graph f(x) = |ln (x + 1)|• The graph is a translation 1
unit to the left. Then the absolute value has the effect of reflecting negative outputs across the x-axis.
• The domain is the set of all real numbers greater than 1.
• The line x = 1 is the vertical asymptote.
1.9466
1.3863
0.6931
00
0.6930.5
f(x)x
Application: Walking Speed
• In a study by psychologists Bornstein and Bornstein, it was found that the average walking speed w, in feet per second, of a person living in a city of population P, in thousands, is given by the function
w(P) = 0.37 ln P + 0.05.
Application: Walking Speed continued
The population of Philadelphia, Pennsylvania, is 1,517,600. Find the average walking speed of people living
in Philadelphia.
Since 1,517,600 = 1517.6 thousand, we substitute 1517.6 for P, since P is in
thousands:
w(1517.6) = 0.37 ln 1517.6 + 0.05 2.8 ft/sec.
The average walking speed of people living in Philadelphia is about 2.8 ft/sec.