3
Exponential and Logarithmic
Functions
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3.3
Properties of Logarithms
Copyright © Cengage Learning. All rights reserved.
What You Should Learn
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Rewrite logarithms with different bases.
Use properties of logarithms to evaluate or
rewrite logarithmic expressions.
Use properties of logarithms to expand or
condense logarithmic expressions.
Use logarithmic functions to model and solve
real-life problems.
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Change of Base
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Change of Base
Most calculators have only two types of log keys, one for
common logarithms (base 10) and one for natural
logarithms (base e).
Although common logs and natural logs are the most
frequently used, you may occasionally need to evaluate
logarithms to other bases. To do this, you can use the
following change-of-base formula.
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Change of Base
One way to look at the change-of-base formula is that
logarithms to base a are simply constant multiples of
logarithms to base b. The constant multiplier is
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Example 1 – Changing Bases Using Common Logarithms
a. Log425
Use a Calculator.
 2.23
Simplify.
b. Log212
 3.58
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Properties of Logarithms
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Properties of Logarithms
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Example 3 – Using Properties of Logarithms
Write each logarithm in terms of ln 2 and ln 3.
a. ln 6
b. ln
Solution:
a. ln 6 = ln(2  3)
b. ln
Rewrite 6 as 2  3.
= ln 2 + ln 3
Product Property
= ln 2 – ln 27
Quotient Property
= ln 2 – ln 33
Rewrite 27 as 33
= ln 2 – 3 ln 3
Power Property
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Rewriting Logarithmic Expressions
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Rewriting Logarithmic Expressions
The properties of logarithms are useful for rewriting
logarithmic expressions in forms that simplify the
operations of algebra. This is true because they convert
complicated products, quotients, and exponential forms into
simpler sums, differences, and products, respectively.
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Example 5 – Expanding Logarithmic Expressions
Use the properties of logarithms to expand each
expression.
a. log45x3y
b. ln
Solution:
a. log45x3y = log45 + log4x3 + log4 y
= log45 + 3 log4x + log4y
Product Property
Power Property
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Example 5 – Solution
cont’d
Rewrite radical using
rational exponent.
Quotient Property
Power Property
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Rewriting Logarithmic Expressions
In Example 5, the properties of logarithms were used to
expand logarithmic expressions.
In Example 6, this procedure is reversed and the properties
of logarithms are used to condense logarithmic
expressions.
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Example 6 – Condensing Logarithmic Expressions
Use the properties of logarithms to condense each
expression.
a.
log10x + 3 log10(x + 1)
b. 2ln(x + 2) – lnx
c. [log2x + log2(x – 4)]
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Example 6 – Solution
a.
log10x + 3 log10(x + 1) = log10x1/2 + log10(x + 1)3
Power Property
Product Property
b. 2 ln(x + 2) – ln x = ln(x + 2)2 – ln x
Power Property
Quotient Property
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Example 6 – Solution
c.
[log2x + log2(x – 4)] = {log2[x(x – 4)]}
= log2[x(x – 4)]1/3
cont’d
Product Property
Power Property
Rewrite with a
radical.
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Application
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Example 7 – Finding a Mathematical Model
The table shows the mean distance x from the sun and the
period y (the time it takes a planet to orbit the sun) for each
of the six planets that are closest to the sun. In the table, the
mean distance is given in astronomical units (where the
Earth’s mean distance is defined as 1.0), and the period is
given in years.
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Example 7 – Finding a Mathematical Model
cont’d
The points in the table are plotted in Figure 3.22. Find an
equation that relates y and x.
Figure 3.22
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Example 7 – Solution
From Figure 3.22, it is not clear how to find an equation that
relates y and x.
To solve this problem, take the natural log of each of the xand y-values in the table. This produces the following
results.
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Example 7 – Solution
cont’d
Now, by plotting the points in the table,
you can see that all six of the points
appear to lie in a line, as shown
in Figure 3.23.
Figure 3.23
To find an equation of the line through these points, you can
use algebraic method.
Choose any two points to determine the slope of the line.
Using the two points (0.421, 0.632) and (0, 0) you can
determine that the slope of the line is
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Example 7 – Solution
cont’d
By the point-slope form, the equation of the line is
Where Y = ln y and X = ln x .You can therefore conclude
that
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