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The Humongous Book of Algebra Problems by W. Michael Kelley

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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 cant
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.

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