10
THE NATURE OF GROWTH
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10.1
Exponential Equations
Copyright © Cengage Learning. All rights reserved.
Exponential Equations
The measurement of growth and decay often involves the
study of relatively large or relatively small quantities.
Difficulty with scaling measurements is often one of our
primary concerns when describing and measuring figures
and data.
Large and small numbers are often represented in
exponential form and scientific notation.
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Exponential Equations
In this section, we investigate solving equations known as
exponential equations
An exponential is an expression of the form bx; The
number b is called the base. In definition bx is defined as an
integer.
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Exponential Equations
Let’s solve the exponential equation 2x = 14. To solve an
equation means to find the replacement(s) for the variable
that make the equation true.
You might try certain values:
x = 1:
x = 2:
x = 3:
x = 4:
2x = 21 = 2
2x = 22 = 4
2x = 23 = 8
2x = 24 = 16
Too small
Too small
Still too small
Too big
It seems as if the number you are looking for is between
3 and 4. Our task in this section is to find both an
approximate as well as an exact value for x.
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Definition of Logarithm
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Definition of Logarithm
The solution of the equation 2x = 14 seeks an x-value.
What is this x-value?
We express the idea in words:
x is the exponent on a base 2 that gives the answer 14
This can be abbreviated as
x = exp on base 2 to give 14
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Definition of Logarithm
We further shorten this notation to
x = exp214
This statement is read, “x is the exponent on a base 2 that
gives the answer 14.”
It appears that the equation is now solved for x, but this is
simply a notational change.
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Definition of Logarithm
The expression “exponent of 14 to the base 2” is called, for
historical reasons, “the log of 14 to the base 2.”
That is,
x = exp214
and
x = log214
mean exactly the same thing.
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Definition of Logarithm
This leads us to the following definition of logarithm.
The statement x = logb A should be read as “x is the log
(exponent) on a base b that gives the value A.”
Do not forget that a logarithm is an exponent.
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Example 2 – Write exponentials in logarithmic form
Write in logarithmic form:
a. 52 = 25
Solution:
a. In 52 = 25, 5 is the base and 2 is the exponent, so we
write
2 = log5 25
Remember, the logarithmic expression “solves” for the
exponent.
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Example 2 – Solution
b. With
cont’d
= 2–3, the base is 2 and the exponent is –3:
–3 = log2
c. With
(since
= 8, the base is 64 and the exponent is
= 641/2 ):
= log64 8
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Definition of Logarithm
In elementary work, the most commonly used base is 10,
so we call a logarithm to the base 10 a common
logarithm, and we agree to write it without using a
subscript 10.
That is, log x is a common logarithm. A logarithm to the
base e is called a natural logarithm and is denoted by
ln x.
The expression ln x is often pronounced “ell en x” or “lon x.”
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Definition of Logarithm
The solution for the equation 10x = 2 is x = log 2, and the
solution for the equation ex = 0.56 is x = ln 0.56.
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Evaluating Logarithms
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Evaluating Logarithms
To evaluate a logarithm means to find a numerical value
for the given logarithm.
To evaluate a logarithm to some base other than base 10
or base e.
The first method uses the definition of logarithm that is, For
positive b and A, b ≠ 1, x = logb A means bx = A. x is called
the logarithm and A is called the argument.
The second method uses what is called the change of
base theorem. Before we state this theorem, we consider
its plausibility with the next example.
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Example 6 – Evaluate logarithmic expressions
Evaluate the given expression.
Solution:
a. From the definition of logarithm, log2 8 = x means 2x = 8
or x = 3. Thus, log2 8 = 3. By calculator,
Also,
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Evaluating Logarithms
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Example 7 – Evaluate logarithms with a change of base
Evaluate (round to the nearest hundredth):
a. log7 3
b. log3 3.84
Solution:
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Evaluating Logarithms
We now return to the problem of solving
2x = 14
Given equation.
x = log2 14
Solution
We call log2 14 the exact solution for the equation, and
Example 8 finds an approximate solution.
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Example 8 – Solve an exponential equation
Solve 2x = 14 (correct to the nearest hundredth).
Solution:
We use the definition of logarithm and the change of base
theorem to write
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Exponential Equations
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Exponential Equations
We now turn to solving exponential equations. Exponential
equations will fall into one of three types:
The next example illustrates the procedure for solving each
type of exponential equation.
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Example 9 – Solve exponential equations with common and natural logs
Solve the following exponential equations:
a. 10x = 5
b. e–0.06x = 3.456
c. 8x = 156.8
Solution:
Regardless of the base, we use the definition of logarithm
to solve an exponential equation.
a. 10x = 5
x = log 5
 0.6989700043
Given equation
Definition of logarithm this is the exact answer.
Approximate calculator answer
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Example 9 – Solution
b.
e–0.06x = 3.456
–0.06x = ln 3.456
cont’d
Given equation
Definition of logarithm
Exact answer; this can be simplified to
in 3.456.
 –20.66853085
Approximate calculator answer
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Example 9 – Solution
c. 8x = 156.8
x = log8 156.8
 2.43092725
cont’d
Given equation
Definition of logarithm; this is the exact answer
Approximate calculator answer. Use the
change of base theorem:
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Example 9 – Solution
cont’d
Note:
Many people will solve this by “taking the log of both sides”:
8x = 156.8
log 8x = log 156.8
Given equation
Take the “log of both the sides”
x log 8 = log 156.8
Divide both sides by log 8.
 2.43092725
This answer agrees with the first solution.
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Exponential Equations
Did you notice that the results of these two calculations in
part c are the same?
It simply involves several extra steps and some additional
properties of logarithms.
It is rather like solving quadratic equations by completing
the square each time instead of using the quadratic
formula.
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Exponential Equations
You can see that, before calculators, there were good
reasons to avoid representations such as log8 156.8.
Whenever you see an expression such as log8 156.8, you
know how to calculate it:
log 156.8/log 8.
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