The Humongous Book of Algebra Problems

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 doesn’t

repeat, like π. You

don’t 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... .