5. Inverse, Exponential and
Logarithmic Functions
5.4 Logarithmic Function
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Logarithmic Functions
The exponential function given by for f(x) = ax for 0 < a < 1
or a > 1 is one-to-one. Hence, f has an inverse function f –1.
This inverse of the exponential function with base a is
called the logarithmic function with base a and is
denoted by loga.
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Logarithmic Functions
Its values are written loga (x) or loga x, read “the logarithm
of x with base a.” Since, by the definition of an inverse
function f –1,
y = f –1(x)
if and only if
x = f(y),
the definition of loga may be expressed as follows.
Note that the two equations in the definition are equivalent.
We call the first equation the logarithmic form and the
second the exponential form.
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Logarithmic Functions
You should strive to become an expert in changing each
form into the other.
The following diagram may help you achieve this goal.
Observe that when forms are changed, the bases of the
logarithmic and exponential forms are the same.
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Logarithmic Functions
The number y (that is, loga x) corresponds to the exponent
in the exponential form.
In words, loga x is the exponent to which the base a must
be raised to obtain x.
This is what people are referring to when they say
“Logarithms are exponents.”
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Logarithmic Functions
The following illustration contains examples of equivalent
forms.
Illustration: Equivalent Forms
Logarithmic form
Exponential form
• log5 u = 2
52 = u
• logb 8 = 3
b3 = 8
• r = logp q
pr = q
• w = log4 (2t + 3)
4w = 2t + 3
• log3 x = 5 + 2z
35 + 2z = x
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Logarithmic Functions
The next example contains an application that involves
changing from an exponential form to a logarithmic form.
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Example 1 – Changing exponential form to logarithmic form
The number N of bacteria in a certain culture after t hours
is given by N = (1000)2t. Express t as a logarithmic function
of N with base 2.
Solution:
N = (1000)2t
given
isolate the exponential expression
change to logarithmic form
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Logarithmic Functions
The following general properties follow from the
interpretation of loga x as an exponent.
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Logarithmic Functions
The reason for property 4 follows directly from the definition
of loga, since
if
y = loga x,
then
x = ay,
or
The logarithmic function with base a is the inverse of the
exponential function with base a, so the graph of y = loga x
can be obtained by reflecting the graph of y = ax through
the line y = x.
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Logarithmic Functions
This procedure is illustrated in Figure 1 for the case a > 1.
Figure 1
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Logarithmic Functions
Note that the x-intercept of the graph is 1, the domain is the
set of positive real numbers, the range is , and the y-axis
is a vertical asymptote.
Logarithms with base 0 < a < 1 are seldom used, so we will
not emphasize their graphs.
We see from Figure 1 that if a > 1, then loga x is increasing
on (0, ) and hence is one-to-one.
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Logarithmic Functions
When using this theorem as a reason for a step in the
solution to an example, we will state that logarithmic
functions are one-to-one.
In the following example we solve a simple logarithmic
equation—that is, an equation involving a logarithm of an
expression that contains a variable.
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Logarithmic Functions
Extraneous solutions may be introduced when logarithmic
equations are solved.
Hence, we must check solutions of logarithmic equations to
make sure that we are taking logarithms of only positive
real numbers; otherwise, a logarithmic function is not
defined.
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Example 3 – Solving a logarithmic equation
Solve the equation log6 (4x – 5) = log6 (2x + 1).
Solution:
log6 (4x – 5) = log6 (2x + 1)
(4x – 5) = (2x + 1)
2x = 6
x=3
given
logarithmic functions are one-to-one
subtract 2x; add 5
divide by 2
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Example 3 – Solution
cont’d
Check x = 3 LS: log6 (4  3 – 5) = log6 7
RS: log6 (2  3 + 1) = log6 7
Since log6 7 = log6 7 is a true statement, x = 3 is a solution.
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Logarithmic Functions
When we check the solution x = 3 in Example 3, it is not
required that the solution be positive.
But it is required that the two expressions, 4x – 5 and
2x + 1, be positive after we substitute 3 for x.
If we extend our idea of argument from variables to
expressions, then when checking solutions, we can simply
remember that arguments must be positive.
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Example 5 – Sketching the graph of a logarithmic function
Sketch the graph of f if f(x) = log3 x.
Solution:
We will describe three methods for sketching the graph.
Method 1:
Since the functions given by log3 x
and 3x are inverses of each other,
we proceed as we did for y = loga x in
Figure 1; that is, we first sketch the
graph of y = 3x and then reflect it
through the line y = x.
Figure 1
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Example 5 – Solution
cont’d
This gives us the sketch in Figure 2. Note that the points
(–1, 3–1), (0, 1), (1, 3), and (2, 9) on the graph of y = 3x
reflect into the points (3–1, –1), (1, 0), (3, 1), and (9, 2) on
the graph of y = log3 x.
Figure 2
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Example 5 – Solution
cont’d
Method 2:
We can find points on the graph of y = log3 x by letting
x = 3k, where k is a real number,
y = log3 x = log3 3k = k
Using this formula, we obtain the points on the graph listed
in the following table.
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Example 5 – Solution
cont’d
This gives us the same points obtained using the first
method.
Method 3:
We can sketch the graph of y = log3 x by sketching the
graph of the equivalent exponential form x = 3y.
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Logarithmic Functions
Logarithms with base 10 are called common logarithms.
The symbol log x is used as an abbreviation for log10 x, just
as
is used as an abbreviation for
.
The natural exponential function is given by f(x) = ex.
The logarithmic function with base e is called the natural
logarithmic function.
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Logarithmic Functions
The symbol ln x (read “ell-en of x”) is an abbreviation for
loge x, and we refer to it as the natural logarithm of x.
Thus, the natural logarithmic function and the natural
exponential function are inverse functions of each other.
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Example 10 – Solving a simple logarithmic equation
Find x if
(a) log x = 1.7959
(b) ln x = 4.7
Solution:
(a) Changing log x = 1.7959 to its equivalent exponential
form gives us
x = 101.7959.
Evaluating the last expression to three-decimal-place
accuracy yields
x  62.503.
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Example 10 – Solution
cont’d
(b) Changing ln x = 4.7 to its equivalent exponential form
gives us
x = e4.7  109.95.
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Logarithmic Functions
The following chart lists common and natural logarithmic
forms
The last property for natural logarithms allows us to write
the number a as eln a, so the exponential function f(x) = ax
can be written as f(x) = (eln a)x or as f(x) = ex ln a. Many
calculators compute an exponential regression model of
the form y = abx.
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Logarithmic Functions
If an exponential model with base e is desired, we can write
the model
y = abx
as
y = aex ln b.
Illustration: Converting to Base e Expressions
• 3x
is equivalent to ex ln 3
• x3
is equivalent to
e3 ln x
• 4  2x
is equivalent to
4  ex ln 2
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Logarithmic Functions
Figure 9 shows four logarithm graphs with base a > 1. Note
that for x > 1, as the base of the logarithm increases, the
graphs increase more slowly (they are more horizontal).
Figure 9
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Logarithmic Functions
This makes sense when we consider the graphs of the
inverses of these functions: y = 2x, y = ex, y = 3x, and
y = 10x.
Here, for x > 0, as the base of the exponential increases,
the graphs increase faster (they are more vertical).
The next example illustrate application of common and
natural logarithms.
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Example 11 – The Richter scale
On the Richter scale, the magnitude R of an earthquake of
intensity I is given by
where I0 is a certain minimum intensity.
(a) If the intensity of an earthquake is 1000I0, find R.
(b) Express I in terms of R and I0.
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Example 11 – Solution
(a)
given
let I = 1000I0
= log 1000
cancel l0
= log 103
1000 = 103
=3
log 10x = x for every x
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Example 11 – Solution
cont’d
From this result we see that a tenfold increase in intensity
results in an increase of 1 in magnitude (if 1000 were
changed to 10,000, then 3 would change to 4).
(b)
given
change to exponential form
multiply by l0
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5. Inverse, Exponential and
Logarithmic Functions
5.4 Logarithmic Function
Copyright © Cengage Learning. All rights reserved.
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