3.3 Properties of Logarithms - Central Bucks School District
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Section 3.3 Properties of Logarithms
239
3.3 Properties of Logarithms
What you should learn
? Use the change-of-base formula to rewrite and evaluate logarithmic expressions.
? Use properties of logarithms to evaluate or rewrite logarithmic expressions.
? Use properties of logarithms to expand or condense logarithmic expressions.
? Use logarithmic functions to model and solve real-life problems.
Why you should learn it
Logarithmic functions can be used to model and solve real-life problems. For instance, in Exercises 81?83 on page 244, a logarithmic function is used to model the relationship between the number of decibels and the intensity of a sound.
AP Photo/Stephen Chernin
Change of Base
Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base e). Although common logs and natural logs are the most frequently used, you may occasionally need to evaluate logarithms to other bases. To do this, you can use the following change-of-base formula.
Change-of-Base Formula
Let a, b, and x be positive real numbers such that a 1 and b 1. Then loga x can be converted to a different base as follows.
Base b
Base 10
Base e
loga
x
logb logb
x a
loga
x
log log
x a
loga
x
ln ln
x a
One way to look at the change-of-base formula is that logarithms to base a are simply constant multiples of logarithms to base b. The constant multiplier is 1logb a.
Example 1 Changing Bases Using Common Logarithms
a.
log4 25
log 25 log 4
1.39794 0.60206
2.3219
log x loga x log a Use a calculator. Simplify.
log 12 1.07918
b.
log2 12
log 2
3.5850
0.30103
Now try Exercise 1(a).
Example 2 Changing Bases Using Natural Logarithms
a.
log4 25
ln 25 ln 4
3.21888 1.38629
2.3219
loga
x
ln ln
x a
Use a calculator. Simplify.
b.
log2 12
ln 12 ln 2
2.48491 0.69315
3.5850
Now try Exercise 1(b).
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240
Chapter 3 Exponential and Logarithmic Functions
Encourage your students to know these properties well. They will be used for solving logarithmic and exponential equations, as well as in calculus.
Properties of Logarithms
You know from the preceding section that the logarithmic function with base a is the inverse function of the exponential function with base a. So, it makes sense that the properties of exponents should have corresponding properties involving logarithms. For instance, the exponential property a0 1 has the corresponding logarithmic property loga 1 0 .
There is no general property that can be used to rewrite logau ? v. Specifically, logau v is not equal to loga u loga v.
Remind your students to note the domain when applying properties of logarithms to a logarithmic function. For example, the domain of f x ln x2 is all real numbers x 0, whereas the domain of gx 2 ln x is all real numbers x > 0.
Properties of Logarithms
Let a be a positive number such that a 1, and let n be a real number. If u and v are positive real numbers, the following properties are true.
Logarithm with Base a
Natural Logarithm
1. Product Property: logauv loga u loga v lnuv ln u ln v
2.
Quotient Property:
loga
u v
loga
u loga v
ln u ln u ln v v
3. Power Property: loga un n loga u
ln un n ln u
For proofs of the properties listed above, see Proofs in Mathematics on page 278.
Example 3 Using Properties of Logarithms
The Granger Collection
Historical Note John Napier, a Scottish mathematician, developed logarithms as a way to simplify some of the tedious calculations of his day. Beginning in 1594, Napier worked about 20 years on the invention of logarithms. Napier was only partially successful in his quest to simplify tedious calculations. Nonetheless, the development of logarithms was a step forward and received immediate recognition.
Write each logarithm in terms of ln 2 and ln 3.
a. ln 6 Solution
2 b. ln
27
a. ln 6 ln2 3
Rewrite 6 as 2 3.
ln 2 ln 3
Product Property
b. ln 2 ln 2 ln 27 27
Quotient Property
ln 2 ln 33
Rewrite 27 as 33.
ln 2 3 ln 3
Power Property
Now try Exercise 17.
Example 4 Using Properties of Logarithms
Find the exact value of each expression without using a calculator.
a. log5 3 5 b. ln e6 ln e2
Solution
a.
log5 3 5
log5
513
1 3
log5
5
1 3
1
1 3
b.
ln
e6
ln
e2
ln
e6 e2
ln e4
4 ln e 41 4
Now try Exercise 23.
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Section 3.3 Properties of Logarithms
241
Rewriting Logarithmic Expressions
The properties of logarithms are useful for rewriting logarithmic expressions in forms that simplify the operations of algebra. This is true because these properties convert complicated products, quotients, and exponential forms into simpler sums, differences, and products, respectively.
A common error made in expanding logarithmic expressions is to rewrite log axn as n log ax instead of as log a n log x.
Exploration
Use a graphing utility to graph the functions given by
y1 ln x lnx 3 and
y2
ln
x
x
3
in the same viewing window. Does the graphing utility show the functions with the same domain? If so, should it? Explain your reasoning.
Example 5 Expanding Logarithmic Expressions
Expand each logarithmic expression.
a. log4 5x3y
3x 5 b. ln
7
Solution
a. log4 5x3y log4 5 log4 x3 log4 y
log4 5 3 log4 x log4 y
b.
3x 5 ln
ln 3x 512
7
7
ln3x 512 ln 7 1 ln3x 5 ln 7
2
Product Property Power Property
Rewrite using rational exponent. Quotient Property
Power Property
Now try Exercise 47.
In Example 5, the properties of logarithms were used to expand logarithmic expressions. In Example 6, this procedure is reversed and the properties of logarithms are used to condense logarithmic expressions.
A common error made in condensing logarithmic expressions is to rewrite
log
x
log
y
as
log log
x y
instead
of
as
log
x . y
Example 6 Condensing Logarithmic Expressions
Condense each logarithmic expression.
a.
1 2
log
x
3
logx
1
c.
1 3
log2
x
log2x
1
b. 2 lnx 2 ln x
Solution
a.
1 2
log
x
3
logx
1
log
x12
logx
13
logxx 13
b. 2 lnx 2 ln x lnx 22 ln x
x 22 ln
x
c.
1 3
log2
x
log2x
1
1 3
log2xx
1
log2xx 113
log2 3 xx 1
Now try Exercise 69.
Power Property Product Property Power Property
Quotient Property
Product Property Power Property Rewrite with a radical.
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Chapter 3 Exponential and Logarithmic Functions
Application
One method of determining how the x- and y-values for a set of nonlinear data are related is to take the natural logarithm of each of the x- and y-values. If the points are graphed and fall on a line, then you can determine that the x- and y-values are related by the equation
ln y m ln x
where m is the slope of the line.
Example 7 Finding a Mathematical Model
y Planets Near the Sun
The table shows the mean distance x and the period (the time it takes a planet to orbit the sun) y for each of the six planets that are closest to the sun. In the table, the mean distance is given in terms of astronomical units (where Earth's mean distance is defined as 1.0), and the period is given in years. Find an equation that relates y and x.
30
Saturn
25
Planet
Mean distance, x
Period, y
Period (in years)
20
Mercury
15
Venus
10
Earth
Jupiter
5
Mars
x
2 4 6 8 10
Mean distance
(in astronomical units)
Solution
Mercury Venus Earth Mars Jupiter Saturn
0.387 0.723 1.000 1.524 5.203 9.537
0.241 0.615 1.000 1.881 11.863 29.447
FIGURE 3.23
The points in the table above are plotted in Figure 3.23. From this figure it is not clear how to find an equation that relates y and x. To solve this problem, take the natural logarithm of each of the x- and y-values in the table. This produces the following results.
ln y
Saturn
3
Jupiter
2
1
Earth Venus
ln y = 3 ln x
2
Mars
ln x
1
2
3
Mercury FIGURE 3.24
Planet ln x ln y
Mercury 0.949 1.423
Venus 0.324 0.486
Earth 0.000 0.000
Mars 0.421 0.632
Jupiter 1.649 2.473
Saturn 2.255 3.383
Now, by plotting the points in the second table, you can see that all six of the points appear to lie in a line (see Figure 3.24). Choose any two points to determine the slope of the line. Using the two points 0.421, 0.632 and 0, 0, you can determine that the slope of the line is
m
0.632 0.421
0 0
1.5
3 .
2
By
the
point-slope
form,
the
equation
of
the
line
is
Y
3 2
X,
where
Y
ln
y
and
X
ln
x.
You
can
therefore
conclude
that
ln
y
3 2
ln
x.
Now try Exercise 85.
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3.3 Exercises
Section 3.3 Properties of Logarithms
243
VOCABULARY CHECK: In Exercises 1 and 2, fill in the blanks.
1. To evaluate a logarithm to any base, you can use the ________ formula. 2. The change-of-base formula for base e is given by loga x ________. In Exercises 3?5, match the property of logarithms with its name.
3. logauv loga u loga v
(a) Power Property
4. ln un n ln u
(b) Quotient Property
5.
loga
u v
loga
u
loga
v
(c) Product Property
PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at .
In Exercises 1?8, rewrite the logarithm as a ratio of (a) common logarithms and (b) natural logarithms.
1. log5 x
3. log15 x
5.
logx
3 10
7. log2.6 x
2. log3 x
4. log13 x
6.
logx
3 4
8. log7.1 x
In Exercises 9?16, evaluate the logarithm using the change-of-base formula. Round your result to three decimal places.
9. log3 7 11. log12 4 13. log9 0.4 15. log15 1250
10. log7 4 12. log14 5 14. log20 0.125 16. log3 0.015
In Exercises 17?22, use the properties of logarithms to rewrite and simplify the logarithmic expression.
17. log4 8
19.
log5
1 250
21. ln5e6
18. log242 34
20.
log
9 300
6 22. ln e2
In Exercises 23?38, find the exact value of the logarithmic expression without using a calculator. (If this is not possible, state the reason.)
23. log3 9 25. log24 8 27. log4 161.2 29. log39 31. ln e4.5
24.
log5
1 125
26. log6 3 6
28. log3 810.2
30. log216
32. 3 ln e4 33. ln 1
e 34. ln 4 e3 35. ln e2 ln e5 36. 2 ln e6 ln e5 37. log5 75 log5 3 38. log4 2 log4 32
In Exercises 39?60, use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)
39. log4 5x
41. log8 x4
5 43. log5 x 45. ln z 47. ln xyz2
49. ln zz 12, z > 1
a 1 51. log2 9 , a > 1
53. ln 3 x y x 4y 55. ln z5
x2 57. log5 y2z3 59. ln 4 x3x2 3
40. log3 10z y
42. log10 2 1
44. log6 z3 46. ln3 t 48. log 4x2 y
x2 1
50. ln x3 , x > 1 6
52. ln x2 1
x2
54. ln y3 x y4
56. log2 z4 xy4
58. log10 z5 60. ln x2x 2
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