Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
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
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:
= 3(log y)
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
10 = log
2 + log
words, the log of a
fraction equals the
log of the numerator
minus the log of the