
Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
407
Note: Problems 18.17–18.20 refer to the function f(x) = log x.
18.18 Evaluate f(12) using a calculator, and round the answer to the thousandths
place.
Different calculators provide different default degrees of decimal accuracy, but
all scientific and graphic calculators should report at least three or four places
behind the decimal.
log 12 ≈ 1.079
Note: Problems 18.17–18.20 refer to the function f(x) = log x.
18.19 Solve the equation for x: log x = –1.
The equation contains the common logarithm log x (with no explicitly stated
base). Therefore, the implied base of the logarithm is 10.
log
10
x = –1
Express the logarithmic equation as an exponential equation and solve it for x.
Note: Problems 18.17–18.20 refer to the function f(x) = log x.
18.20 Identify consecutive integers a and b such that a < log 781 < b.
The expression log 781 is a common logarithm, with implied base 10:
log 781 = log
10
781. The expression log
10
781 has the same value as the exponent
to which 10 must be raised to produce a result of 781. Consider the following
powers of 10: 10
2
= 100 and 10
3
= 1,000. Therefore, log 100 = 2 and log 1,000 = 3.
The domain of f(x) is all positive real numbers, and the graph of f(x) increases
monotonically over its entire domain; hence log 100 < log 781 < log 1,000 and
2 < log 781 < 3. Therefore, a = 2 and b = 3.
In other words,
round the answer
to three decimal
places.
The actual decimal is more like
1.07918124604762482772…. that
means 10
1.07918124604762482772…
= 12.
If you trace
the graph of f(x)
from left to right,
you notice that the
graph always goes up.
That means the bigger
the x-value, the bigger
the value of log x.
Therefore, log 781 is
bigger than log 100
(which equals 2) but
smaller than log
1,000 (which
equals 3).