=
8.3
=
log 10
8.3
I
.
10
8.3
I
0
R
=
log
10
8.3
I
0
I
0
R
=
log
I
I
0
10
8.3
I
0
.
I
,
10
8.3
10
8.3
I
0
R
=
log
I
I
0
,
I
R
,
EXAMPLE 9
Use natural logarithms.
[
−
10, 10, 1] by [
−
10, 10, 1]
Figure 3.14
The domain of
is
1
 q
, 3
2
.
f
1
x
2
=
ln
1
3

x
2
[
−
10, 10, 1] by [
−
10, 10, 1]
Figure 3.15
3 is excluded from
the domain of
h
1
x
2
=
ln
1
x

3
2
2
.
408
Chapter 3
Exponential and Logarithmic Functions
Section 3.2
Logarithmic Functions
409
The basic properties of logarithms that were listed earlier in this section can be
applied to natural logarithms.
Properties of Natural Logarithms
Inverse
properties
General Properties
Natural Logarithms
log
b
1=0
1.
log
b
b=1
2.
log
b
b
x
=x
3.
b
log
b
x
=x
4.
ln
1=0
1.
ln
e=1
2.
ln
e
x
=x
3.
e
ln
x
=x
4.
Examine the inverse properties,
and
Can you see how ln
and
“undo” one another? For example,
Dangerous Heat: Temperature in an Enclosed Vehicle
When the outside air temperature is anywhere from 72° to 96° Fahrenheit, the
temperature in an enclosed vehicle climbs by 43° in the first hour. The bar graph in
Figure 3.16
shows the temperature increase throughout the hour.The function
models the temperature increase,
in degrees Fahrenheit, after
minutes. Use
the function to find the temperature increase, to the nearest degree, after 50 minutes.
How well does the function model the actual increase shown in
Figure 3.16
?
x
f
1
x
2
,
f
1
x
2
=
13.4 ln
x

11.6
EXAMPLE 11
ln
e
2
=
2,
ln
e
7
x
2
=
7
x
2
,
e
ln 2
=
2,
and
e
ln 7
x
2
=
7
x
2
.
e
e
ln
x
=
x
.
ln
e
x
=
x
60
43
°
50
41
°
40
38
°
30
34
°
20
29
°
45
°
40
°
35
°
30
°
25
°
20
°
15
°
10
°
Temperature Increase (
°
F)
Temperature Increase
in an Enclosed Vehicle
Minutes
10
19
°
5
°
Figure 3.16
Source:
Professor Jan Null, San Francisco State
University
Solution
We find the temperature increase after 50 minutes by substituting
50 for
and evaluating the function at 50.
This is the given function.
Substitute 50 for
Graphing calculator keystrokes:
On some
calculators, a parenthesis is needed after 50.
According to the function, the temperature will increase by approximately 41° after
50 minutes. Because the increase shown in
Figure 3.16
is 41°, the function models the
actual increase extremely well.
Check Point
11
Use the function in Example 11 to find the temperature
increase, to the nearest degree, after 30 minutes. How well does the function
model the actual increase shown in
Figure 3.16
?
13.4
ln
50

11.6
ENTER
.
L
41
x
.
f
1
50
2
=
13.4 ln 50

11.6
f
1
x
2
=
13.4 ln
x

11.6
x
The Curious Number
e
You will learn more about each curiosity mentioned below when you take calculus.
•
The number
was named by the Swiss mathematician Leonhard Euler (1707–1783),
who proved that it is the limit as
of
•
features in Euler’s remarkable relationship
in which
•
The first few decimal places of
are fairly easy to remember:
•
The best rational approximation of
using numbers less than 1000 is also easy to
remember:
•
Isaac Newton (1642–1727),
one of the cofounders of calculus,
showed that
from which we obtain
an infinite sum suitable for calculation
because its terms decrease so rapidly. (
Note
:
(
factorial) is the product of all the consecutive integers from
down to 1:
)
•
The area of the region bounded by
the
and
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