A. Solving Equations Using the Fundamental Properties of Logarithms
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Section 5.4 Properties of Logarithms
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Fundamental Properties of Logarithms
For any base b 7 0, b 1,
I. logb b ϭ 1, since b1 ϭ b
II. logb 1 ϭ 0, since b0 ϭ 1
III. logb bx ϭ x, since bx ϭ bx
IV. blogb x ϭ x 1x 7 02 , since logb x ϭ logb x
To see the verification of Property IV more clearly, again note that for
y ϭ logb x, b y ϭ x is the exponential form, and substituting logb x for y yields blogb x ϭ x.
Also note that Properties III and IV demonstrate that y ϭ logb x and y ϭ bx are inverse
functions. In common language, “a base-b logarithm undoes a base-b exponential,”
and “a base-b exponential undoes a base-b logarithm.” For f 1x2 ϭ log b x and
f Ϫ1 1x2 ϭ bx, using a composition verifies the inverse relationship just as in Example 5
from Section 5.1:
1 f ؠf Ϫ1 2 1x2 ϭ f 3 f Ϫ1 1x2 4
1 f Ϫ1 ؠf 21x2 ϭ f Ϫ1 3 f 1x2 4
ϭ logb bx
ϭx
ϭ blogb x
ϭx
These properties can be used to solve basic equations involving logarithms and exponentials. From the uniqueness property for exponents (page 496), note that if logb x ϭ k
, then blogb x ϭ bk, and we say that we have exponentiated both sides.
EXAMPLE 1
ᮣ
Solving Basic Logarithmic Equations
Solve each equation by applying fundamental properties. Answer in exact form and
approximate form using a calculator (round to 1000ths).
a. ln x ϭ 2
b. Ϫ0.52 ϭ log x
Solution
ᮣ
a. ln x ϭ 2
eln x ϭ e2
x ϭ e2
Ϸ 7.389
given
exponentiate both sides
Property IV, exact form
approximate form
b. Ϫ0.52 ϭ log x
10Ϫ0.52 ϭ 10log x
10Ϫ0.52 ϭ x
0.302 Ϸ x
given
exponentiate both sides
Property IV, exact form
approximate form
Now try Exercises 7 through 10 ᮣ
Note that checking the exact solutions by
substitution is a direct application of Property III
(Figure 5.38).
Also, we observe that exponentiating both sides
of the equation produces the same result as simply
writing the original equation in exponential form, and
the process can be viewed in terms of either
approach.
EXAMPLE 2
ᮣ
Figure 5.38
Solving Basic Exponential Equations
Solve each equation by applying fundamental properties. Answer in exact form and
approximate form using a calculator (round to 1000ths).
a. ex ϭ 167
b. 10 x ϭ 8.223
Solution
ᮣ
a.
ex ϭ 167
ln ex ϭ ln 167
x ϭ ln 167
x Ϸ 5.118
given
use natural log
Property III, exact form
approximate form
b.
10x ϭ 8.223
log 10x ϭ log 8.223
x ϭ log 8.223
x Ϸ 0.915
given
use common log
Property III, exact form
approximate form
Now try Exercises 11 through 14 ᮣ
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CHAPTER 5 Exponential and Logarithmic Functions
Similar to our observations from Example 1, taking
the logarithm of both sides produced the same result
as writing the equation in logarithmic form, and the
process can be viewed in terms of either approach.
Also note that here, checking the exact solution by
substitution is a direct application of Property IV
(Figure 5.39).
If an equation has a single logarithmic or exponential term (base 10 or base e), the equation can be
solved by isolating this term and applying one of the
fundamental properties as in Examples 1 and 2.
EXAMPLE 3
ᮣ
Figure 5.39
Solving Exponential Equations
Solve each equation. Write answers in exact form and approximate form to four
decimal places.
a. 10x Ϫ 29 ϭ 51
b. 3exϩ1 Ϫ 5 ϭ 7
Solution
ᮣ
a. 10x Ϫ 29 ϭ 51 given
10x ϭ 80 add 29
Since the left-hand side is base 10, we apply the common logarithm.
log 10x ϭ log 80
x ϭ log 80
Ϸ 1.9031
take the common log of both sides
Property III (exact form)
approximate form
b. 3exϩ1 Ϫ 5 ϭ 7
given
3exϩ1 ϭ 12 add 5
exϩ1 ϭ 4
divide by 3
Since the left-hand side is base e, we apply the natural logarithm.
ln exϩ1 ϭ ln 4
x ϩ 1 ϭ ln 4
x ϭ ln 4 Ϫ 1
Ϸ 0.3863
take the natural log of both sides
Property III
solve for x (exact form)
approximate form
Now try Exercises 15 through 20 ᮣ
Figure 5.40
As an alternative to using the exact form to
check solutions, we can STO (store) the exact result
in storage location X,T,,n (the function variable x) and
simply enter the original equation on the home
screen. The verification for Example 3b is shown in
Figure 5.40.
EXAMPLE 4
ᮣ
Solving Logarithmic Equations
Solve each equation. Write answers in exact form and approximate form to four
decimal places.
a. 2 log 17x2 ϩ 1 ϭ 4
b. Ϫ4 ln 1x ϩ 12 Ϫ 5 ϭ 7
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Section 5.4 Properties of Logarithms
Solution
ᮣ
a. 2 log 17x2 ϩ 1 ϭ 4
2 log 17x2 ϭ 3
3
log 17x2 ϭ
2
3
7x ϭ 102
3
102
xϭ
7
Ϸ 4.5175
given
subtract 1
divide by 2
exponential form
divide by 7 (exact form)
approximate form
b. Ϫ4 ln 1x ϩ 12 Ϫ 5 ϭ 7
Ϫ4 ln 1x ϩ 12 ϭ 12
ln 1x ϩ 12 ϭ Ϫ3
x ϩ 1 ϭ eϪ3
x ϭ eϪ3 Ϫ 1
Ϸ Ϫ0.9502
given
add 5
divide by Ϫ4
exponential form
subtract 1 (exact form)
approximate form
Now try Exercises 21 through 26
A. You’ve just seen how
we can solve logarithmic
equations using the
fundamental properties of
logarithms
519
ᮣ
Figure 5.41
As with other kinds of equations, solutions to logarithmic and exponential equations can be found
10
using the intersection-of-graphs method or the
zeroes method. For Example 4a and the intersection method, enter Y1 ϭ 2 log17X2 ϩ 1 and
10
Y2 ϭ 4 on the Y= screen (from the domain of 0
the function and the expected result, we know
to set a window that includes only Quadrant I).
Using the 2nd TRACE (CALC) 5:intersect option,
0
we press
three times to identify each curve
and bypass the “Guess” option. The calculator then finds the point of intersection and
prints it at the bottom of the screen, and verifies our calculated result.
ENTER
B. The Product, Quotient, and Power Properties of Logarithms
Generally speaking, equation solving involves simplifying the equation, isolating a
variable term on one side, and applying an inverse to solve for the unknown. For logarithmic equations such as log x ϩ log 1x ϩ 32 ϭ 1, we must find a way to combine the
terms on the left, before we can work toward a solution. This requires a further exploration of logarithmic properties.
Due to the close connection between exponents and logarithms, their properties
are very similar. To illustrate, we’ll use terms that can all be written in the form 2x, and
write the equations 8 # 4 ϭ 32, 84 ϭ 2, and 82 ϭ 64 in both exponential form and logarithmic form.
The exponents from a
product are added:
exponential form:
logarithmic form:
The exponents from a
quotient are subtracted:
exponential form:
logarithmic form:
The exponents from a
power are multiplied:
exponential form:
logarithmic form:
23 # 22 ϭ 23ϩ2
log2 18 # 42 ϭ log28 ϩ log24
23
ϭ 23Ϫ2
22
8
log2 a b ϭ log2 8 Ϫ log2 4
4
123 2 2 ϭ 23 2
log2 82 ϭ 2 # log2 8
Each illustration can be generalized and applied with any base b.
#
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CHAPTER 5 Exponential and Logarithmic Functions
WORTHY OF NOTE
Properties of Logarithms
For a more detailed verification of
these properties, see Appendix VI.
Given M, N, and b
Product Property
1 are positive real numbers, and any real number p.
Quotient Property
Power Property
M
logb 1MN2 ϭ logb Mϩlogb N
logba b ϭ logb MϪlogb N logb Mp ϭ plogb M
N
The log of a product
The log of a quotient is
The log of a quantity
is a sum of logarithms.
a difference of logarithms.
to a power is the
power times the log
of the quantity.
ᮣ
CAUTION
It’s very important that you read and understand these properties correctly. In particular:
log M
log M
M
(1) log1M ϩ N2 log M ϩ log N, (2) log a b
, and (3)
log M Ϫ log N.
N
log N
log N
For M ϭ 100 and N ϭ 10, statement (1) would indicate log 110 ϭ 2 ϩ 1X, (2) would
indicate that 1 ϭ 21X, and (3) would indicate that 21 ϭ 2 Ϫ 1X.
In the statement of these properties, it’s worth reminding ourselves that the equal
sign “works both ways,” and we have logb M ϩ logb N ϭ logb 1MN2 . These properties
are often used to write a sum or difference of logarithmic terms as a single expression.
EXAMPLE 5
ᮣ
Rewriting Expressions Using Logarithmic Properties
Use the properties of logarithms to write each expression as a single term.
a. log2 7 ϩ log2 5
b. 2 ln x ϩ ln 1x ϩ 62
c. log 1x ϩ 22 Ϫ log x
Solution
ᮣ
a. log2 7 ϩ log2 5 ϭ log2 17 # 52
ϭ log2 35
b. 2 ln x ϩ ln 1x ϩ 62 ϭ ln x2 ϩ ln 1x ϩ 62
ϭ ln 3x2 1x ϩ 62 4
ϭ ln 3x3 ϩ 6x2 4
xϩ2
c. log 1x ϩ 22 Ϫ log x ϭ log a
b
x
product property
simplify
power property
product property
simplify
quotient property
Now try Exercises 27 through 42 ᮣ
We can verify that these properties produce equivalent results by entering the original
equation as Y1, the result after applying the properties as Y2, and viewing the results of
various inputs on a TABLE screen. Results for Example 5b (power and product properties) and 5c (quotient property) are shown in Figures 5.42 and 5.43 respectively.
Figure 5.42
2 ln x ϩ ln1x ϩ 62 ϭ ln1x3 ϩ 6x2 2
Figure 5.43
log 1x ϩ 22 Ϫ log x ϭ log 1 x
ϩ 2
x 2