Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
409
18.24 Simplify the expression: ln e
4
– ln e
–7
.
Rewrite the expression as a – b, such that a = ln e
4
and b = ln e
–7
. Explicitly state
the bases of the natural logarithms and express the logarithmic equations as
exponential equations.
Evaluate the expression a – b given the above values.
Change of Base Formula
Calculate log values that have weird bases
18.25 Explain how to use the change of base formula to calculate log
b
a. Assume that
b is a positive real number not equal to 1.
According to the logarithmic change of base formula, .
The quotient of common logarithms is equivalent to the quotient of natural
logarithms.
Note: Problems 18.26–18.27 refer to the real number c = log
7
5.
18.26 Calculate c using the common logarithm version of the change of base
formula and round the answer to the thousandths place.
According to the change of base formula, . Therefore,
. Evaluate both logs using a calculator and then calculate the
quotient.
Notice that
ln e
4
= 4 and
ln e
–7
= –7. If you
plug e to some power
into the natural log,
the log and the e
cancel out leaving
behind only the power.
Sound familiar?
Because ln x and e
x
are inverse functions,
plugging one into the
other causes the
functions to cancel
out. More on that in
Problems 19.19–
19.35.
To calculate
any logarithmic
value, divide the log
of the argument (a)
by the log of the base
(b). You can also divide
the natural log of a by
the natural log of b—
you’ll get the same
answer.
Don’t round
anything until you
get the nal answer.
Every time you round
a decimal during a
problem, it makes the
nal answer less
accurate.
Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
410
Note: Problems 18.26–18.27 refer to the real number c = log
7
5.
18.27 Calculate c using the natural logarithm version of the change of base formula
and round the answer to the thousandths place. Demonstrate that the solution
matches the solution to Problem 18.26.
The change of base formula may be applied using common logarithms (as
Problem 18.26 demonstrates) or natural logarithms: . In fact,
you can use any logarithmic base in the formula, as long as you use the same
base in the numerator and denominator.
The quotient of natural logs generates the same value of c as the quotient of
common logs calculated in Problem 18.26.
Note: Problems 18.28–18.29 refer to the function f(x) = log
3
x.
18.28 Evaluate f(81) using the change of base formula.
Substitute x = 81 into the function.
f(81) = log
3
81
Apply the change of base formula. As demonstrated in Problems 18.26–18.27,
the common logarithm and natural logarithm versions of the formula produce
the same value. The common logarithm quotient is presented here.
If you
can use any
base, why stick
with common and
natural logs? Most
calculators have
a LOG button
and an LN
button.
You don’t need
to use the change
of base formula
here. You can use
the technique from
Problems 18.5–18.7.
Chapter Eighteen — Logarithmic Functions
The Humongous Book of Algebra Problems
411
Note: Problems 18.28–18.29 refer to the function f(x) = log
3
x.
18.29 Evaluate f(14) using the change of base formula and round the answer to the
thousandths place.
The natural logarithm version of the change of base formula is presented here.
18.30 Evaluate log
6
2 using the change of base formula and round the answer to the
thousandths place.
The common and natural logarithm versions of the change of base formula
produce the same result; the common logarithm solution is presented here.
18.31 Solve the equation 3
x
= 8 using the change of base formula and round the
answer to the thousandths place.
Express the exponential equation as a logarithmic equation.
log
3
8 = x
Calculate x using the change of base formula.
To transform
3
x
= 8 into a log,
make the base of
the exponent (3) the
base of the log, take
the log of the right
side of the equation
(8), and set it equal
to the power (x).
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