Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
412
18.32 Solve the equation 9
x
= 13 using the change of base formula and round the
answer to the thousandths place.
Express the exponential equation as a logarithmic equation and solve.
18.33 Solve the equation 7
2x
+ 1 = 6 using the change of base formula and round the
answer to the thousandths place.
Isolate 7
2x
on the left side of the equation.
Rewrite the exponential equation as a logarithmic equation.
log
7
5 = 2x
Calculate log
7
5 using the change of base formula.
Multiply both sides of the equation by to solve for x.
Logarithmic Properties
Expanding, contracting, and simplifying log expressions
18.34 Expand the expression: ln (2x).
Logarithmic expressions can be rewritten according to three fundamental
properties. Specifically, a single logarithm can be expressed as multiple
logarithms and vice versa. The first property states that the logarithm of a
product can be expressed as the sum of the logarithms of its factors:
log
a
(xy) = log
a
x + log
a
y.
Or divide
both sides by 2
Which are
presented one at a
time in Problems 18.34,
18.36, and 18.38
Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
413
In this example, the logarithm of the product 2x may be expressed as the sum of
the logarithms of its factors, 2 and x.
ln (2x) = ln 2 + ln x
18.35 Demonstrate the logarithmic property presented in Problem 18.34 by verifying
that ln 10 = ln 2 + ln 5.
Because 2(5) = 10, the natural logarithm of the product (10) should equal the
sum of the natural logarithms of the factors (2 and 5). Verify using a calculator.
18.36 Expand the expression: .
A property of logarithms states that the logarithm of a quotient is equal to the
difference of the logarithms of the dividend and divisor: .
18.37 Demonstrate the logarithmic property presented in Problem 18.36 by
verifying that log 4 = log 12 – log 3.
Because , the logarithm of the quotient (4) is equal to the difference of
the logarithms of the dividend (12) and the divisor (3).
Note: Problems 18.38–18.39 refer to the expression log y
3
.
18.38 Expand the logarithmic expression.
A property of logarithms states that the logarithm of a quantity raised to an
exponent n is equal to the product of n and the logarithm of the base:
log
a
x
n
= n(log
a
x).
log y
3
= 3(log y)
There are
no properties for
the log of a sum. For
example, ln (2 + x) ≠
(ln 2)(ln x). You also can’t
“distribute” the letters
“ln.” For example,
ln (x + 2) ≠ ln x + ln 2.
This is not
only true for
natural logs but
it also works for any
base, as long as all the
bases in the expression
match. In other words,
log 10 = log 2 + log 5
and log
12
10 = log
12
2 + log
2
5.
In other
words, the log of a
fraction equals the
log of the numerator
minus the log of the
denominator.
Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
414
Note: Problems 18.38–18.39 refer to the expression log y
3
.
18.39 Prove that the expansion generated by Problem 18.38 is valid by applying one
of the two other logarithmic properties.
According to Problem 18.34, log
a
(xy) = log
a
x + log
a
y; the logarithm of a product
is equal to the sum of the logarithms of its factors. The property is equally valid
for a product of three factors.
log
a
(xyz) = log
a
x + log
a
y + log
a
z
Rewrite the expression log y
3
by expanding the argument y
3
of the logarithm.
log y
3
= log (y · y · y)
Apply the logarithmic property just described to express log (y · y · y) as a sum.
= log y + log y + log y
Add like terms.
= 3
log y
Therefore, log y
3
= 3
log y.
18.40 Expand and simplify the logarithmic expression: log
2
(2y
4
).
Apply the property described in Problem 18.34 to express the logarithm of a
product as the sum of the logarithms of its factors.
log
2
(2y
4
) = log
2
2 + log
2
y
4
Apply the property presented in Problem 18.38 to express the logarithm of
an argument that is raised to a power as the product of that power and the
logarithm of the argument: log
2
y
4
= 4
log
2
y.
= log
2
2 + 4
log
2
y.
Simplify the expression, noting that log
2
2 = 1.
= 1 + 4
log
2
y
Therefore, log
2
(2y
4
) = 1 + 4
log
2
y.
18.41 Expand the logarithmic expression: .
Express the quotient as a difference of logarithms.
Express each of the products (4x
2
and 5y
6
) as a sum of logarithms.
= (ln 4 + ln x
2
) – (ln 5 + ln y
6
)
The “argument”
of a log is whatever
you’re taking the log
of. In other words, the
argument of log
2
5 is 5.
The three
“log y” expressions
are the most alike of
any like terms—they’re
exactly the same. The
equation log y + log y +
log y = 3(log y) is true
for the same reason
“dog + dog + dog =
3 dogs” is true.
Set log
2
2 =
c and rewrite
the equation using
exponents.
Distribute
the – sign to ln 5
and ln y
6
.
Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
415
The logarithmic arguments x
2
and y
6
contain exponents. Rewrite those
expressions as a product of the power and the logarithm of the base:
ln x
2
= 2
ln
x and ln
y
6
= 6
ln
y.
= ln 4 + 2
ln
x – ln 5 – 6
ln
y
Therefore, .
18.42 Rewrite the logarithmic expression as a single logarithm: log x + 9
log
y.
According to Problem 18.38, n(log
a
x) = log
a
x
n
. Therefore, 9
log y = log y
9
.
log x + 9
log
y = log x + log y
9
Express the sum of the logarithms as the logarithm of a product.
= log (xy
9
)
Therefore, log x + 9
log
y = log (xy
9
).
18.43 Rewrite the logarithmic expression as a single logarithm:
3
log x – 2
log
y – log
z.
Transform coefficients 3 and 2 into exponents of the respective logarithmic
arguments.
3
log x – 2
log
y – log
z = log
x
3
– log
y
2
– log
z
Express the difference (log
x
3
– log
y
2
) as a logarithmic quotient, according to
the property .
Again, express the difference as the logarithm of a quotient.
Therefore, .
Do the
exact opposite
of what you did
in Problems 18.40
and 18.41. “Contract”
the expression into
one log instead of
expanding it into
many logs.
Instead
of pulling the
exponent OUT
of the log as a
coefcient, stick the
coefcient INTO the
expression as an
exponent.
This is the
log property from
Problem 18.34—it’s
just reversed. You’re
changing a sum into
a product instead
of a product into
a sum.
From
Problem 18.36
To divide a
fraction like
by z,
just write z in the
denominator.
Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
416
18.44 Rewrite the logarithmic expression as a single logarithm:
.
Transform coefficients and 4 into exponents of the respective logarithmic
arguments.
Express the difference as the logarithm of a quotient: .
Express the sum as the logarithm of a product.
Therefore, .
Raising some-
thing to the
one-half power is
the same as taking
the square root of it.
See Problems 13.13–
13.20 for more
info.
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