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,
a =x
y
is equivalent to
y = loga x
Logax is an exponent to which the base a
must be raised to give x.
Logarithmic Form
loga x  y  exp onent

base
Argument
(always
positive)
Exponential Form
exp onent

y
a x

ba s e
All a log is . . . is an exponent!
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
Graph: x = 3y
y = log 3 x
x = 3y
y
(x, y)
1
0
(1, 0)
3
1
(3, 1)
9
2
(9, 2)
1/3
1
(1/3, 1)
1/9
2
(1/9, 2)
1/27
3
(1/27, 3)
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.
The inverse of f(x) = a
is given by
f (x) = loga x
-1
x
Asymptotes
• Recall that the horizontal asymptote
of the exponential function y = ax is
the x-axis.
• The vertical asymptote of a
logarithmic function y = loga x is the
y-axis.
Logarithms
log a x  y  x  a
y
A logarithm is an exponent!
• Convert each of the following to a
logarithmic equation.
a) 25 = 5x
b) ew = 30
Example
• Convert each of the following to an
exponential equation.
• a) log7 343 = 3
The logarithm is the exponent.
log7 343 = 3
7 3 = 343
The base remains the same.
b) logb R = 12
Finding Certain Logarithms
• Find each of the following.
a) log2 16
b) log10 1000
c) log16 4
d) log10 0.001
Common Logarithm
Logarithms, base 10, are called common
logarithms.
For all positive numbers x,
logx = log10 x
•Log button on your calculator
is the common log *
Example
• Find each of the following common logarithms on
a calculator.
Round to four decimal places.
a) log 723,456
b) log 0.0000245
c) log (4)
Function Value
Readout
Rounded
log 723,456
5.859412123
5.8594
log 0.0000245
4.610833916
4.6108
log (4)
ERR: nonreal ans
Does not exist
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 723,456
b) ln 0.0000245
c) ln (4)
Function Value
Readout
Rounded
ln 723,456
13.49179501
13.4918
ln 0.0000245
10.61683744
10.6168
ln (4)
ERR: nonreal ans
Does not exist
Changing Logarithmic Bases
• The Change-of-Base Formula
For any logarithmic bases a and b,
and any positive number M,
log a M
log b M 
.
log a 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:
log10 8
log 6 8 
log10 6
 1.1606
Example
We 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
log e 8
log 6 8 
log e 6
ln 8

 1.1606
ln 6
Properties of Logarithms
For a > 0, a  1
1. loga a = 1
2. loga 1 = 0
For the logarithmic base e,
3. ln e = 1
4. ln 1 = 0
Graphs of Logarithmic Functions
• Graph: y = f(x) = log6 x.
– Select y.
– Compute x.
x, or 6 y
1
6
36
216
1/6
1/36
y
0
1
2
3
1
2
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.
x
2.25
2.5
3
4
5
f(x)
1.386
0.693
0
0.693
1.099
Graph f(x)14 = 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.
x
0.1
0.5
1
3
5
f(x)
2.576
2.173
2
1.725
1.598
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.
x
0.5
0
1
3
6
f(x)
0.693
0
0.693
1.386
1.946
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.