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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
406
Note: Problems 18.15–18.16 refer to the function g(x) = log
3
x.
18.16 Graph .
To transform g(x) = log
3
x into , insert –x into the function
(which reflects the graph about the y-axis) and add 1 to the function (which
shifts its graph up one unit). The graph of is presented in Figure 18-4.
Figure 18-4: The graph of is the graph of g(x) = log
3
x reflected
about the y-axis and shifted up one unit.
Common and Natural Logarithms
What the bases equal when no bases are written
Note: Problems 18.17–18.20 refer to the function f(x) = log x.
18.17 Express the logarithmic equation as an exponential equation.
The function log x with no explicitly defined base is classified as a common
logarithm and has an implied base of 10. Therefore, f(x) = log
10
x. Express the
logarithmic equation as an exponential equation.
The function
f(x) = log x answers
this question: “Raising
10 to what power results
in the value x?
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).
Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
408
Note: Problems 18.21–18.23 refer to the function g(x) = ln x.
18.21 Express the logarithmic function as an exponential function.
The common logarithm function, as described in Problems 18.17–18.20, is
written without an explicit base: log x. The natural logarithm function, written
ln x, also has an implied base but is easily distinguishable from the common log
because of its naming convention.
The implied base of a natural logarithm is e, Euler’s number. Therefore,
ln x = log
e
x. Rewrite the equation as an exponential equation.
Note: Problems 18.21–18.23 refer to the function g(x) = ln x.
18.22 Evaluate using a calculator and round the answer to the thousandths
place.
Common and natural logarithms are usually represented by different buttons
on scientific and graphing calculators.
Note: Problems 18.21–18.23 refer to the function g(x) = ln x.
18.23 Evaluate g(e) without using a calculator.
Substitute e into g(x).
g(e) = ln e
Rewrite the natural logarithm, explicitly stating base e.
g(e) = log
e
e
Rewrite the logarithmic equation as an exponential equation and write both
sides of the equation as powers of e.
Natural
logs are written
“ln,” and are the
only logarithmic
function not written
using the “log” notation
like (“log x” or “log
3
9”).
e is one
of those
predened
math decimals
that stretches on
forever and doesnt
repeat, like π. You
dont have to memorize
its value
(e 2.71828182846…)
because exact answers
are written in terms
of e and approximate
decimal answers
are handled by a
calculator (which
has the value of
e programmed
into it).
The actual
value is more like
-1.098612288668109691... .

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