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C ONCEPT
Concept 1. Properties of Logarithms
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Properties of Logarithms
Here you will learn how to use the properties of logarithms to rewrite more complex logarithmic expressions as
sums and differences of simpler expressions, and also to combine multiple expressions into a single logarithmic
expression.
What is the value of the expression log6 (8) + log6 (27)?
Alone, neither of these expressions has an integer value, therefore combining them might seem like a bit of a
challenge. The value of log6 8 is between 1 and 2; the value of log6 27 is also between 1 and 2.
Is there an easier way?
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- James Sousa: The Properties ofLogarithms
Guidance
Previously we defined the logarithmic function as the inverse of an exponential function, and we evaluated log
expressions in order to identify values of these functions. In this lesson we will work with more complicated log
expressions. We will use the properties of logarithms to write a log expression as the sum or difference of several
expressions, or to write several expressions as a single log expression.
Properties of Logarithms
Because a logarithm is an exponent, the properties of logs reflect the properties of exponents.
The basic properties are:
� �
logb (xy) = logb x + logb y logb xy = logb x − logb y logb xn = nlogb x
Expanding Expressions
Using the properties of logs, we can write a log expression as the sum or difference of simpler expressions. Consider
the following examples:
1. log2 8x
� 2=�log2 8 + log2 x = 3 + log2 x
2. log3 x3 = log3 x2 − log3 3 = 2log3 x − 1
Using the log properties in this way is often referred to as "expanding". In the first example, expanding the log
allowed us to simplify, as log2 8 = 3. Similarly, in the second example, we simplified using the log properties, and
the fact that log3 3 = 1.
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Condensing Expressions (Answer to the concept question in the introduction)
To condense a log expression, we will use the same properties we used to expand expressions. Consider the
expression log6 (8) + log6 (27). Individually, neither of these expressions has an integer value. The value of log6
8 is between 1 and 2; the value of log6 27 is also between 1 and 2.
However, if we condense the expression, we get:
log6 (8) + log6 (27) = log6 (8 · 27) = log6 (216) = 3
Example A
Expand each expression:
TABLE 1.1:
a. log5 25x2 y
Solution:
b. log10
� 100x �
9b
a. log5 (25)x2 y = log5 (25) + log5 x2 + log5 y = 2 + 2log5 x + log5 y b.
TABLE 1.2:
log10
� 100x �
= log10 100x − log10 9b
= log10 100 + log10 x − [log10 9 + log10 b]
= 2 + log10 x − log10 9 − log10 b
9b
Example B
Condense the expression:
2log3 x + log3 5x - log3 (x + 1)
Solution:
2log3 x + log3 5x − log3 (x + 1) = log3 x2 + log3 5x − log3 (x + 1)= log3 (x2 (5x)) −
log3 (x + 1)
� 3�
5x
= log3 x+1
Note
� 3 �that not all solutions may be valid, since the argument must be defined. For example, the expression above:
5x
x+1 is undefined if x = -1.
Example C
Condense the expression:
log2 (x2 - 4) - log2 (x + 2)
Solution:
log2 (x2 − 4) − log2 (x + 2)
= log2 (x − 2)
= log2
�
x2 −4
x+2
�
�
�
(x+2)(x−2)
= log2
x+2
Note that the argument of a log must be positive. For example, the expressions in Example ’C’ above are not defined
for x ≤ 2 (which allows us to "cancel" (x+2) without worrying about the condition x�= -2).
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Concept 1. Properties of Logarithms
Vocabulary
Expanding logs refers to the process of splitting a single log into two separate and simpler logs.
Condensing logs refers to the process of combining two individual logs into a single log.
Guided Practice
1) Condense the following expressions into a single logarithm:
log2 a + log2 b + log2 c
2) Condense the expression into a single logarithm:
log4 m + log4 n − 3log4 x
3) Condense the following into a single logarithm:
3log6 x + 2log6 (3x) − log6 (2x3 )
4) Expand the logarithm:
7
log2 ( 5x
)
3x4
Answers
1) To condense the logs, apply the rule as explained in the lesson above: logx y + logx z = logx y · z
log2 a + log2 b + log2 c → log2 a · b · c
2) Recall that logx y − logx z = logx yz
log4 m + log4 n − log4 x → log4 m·n
x
3) Recall that 3logx y = logx y3
3 +3x2
x+3
3log6 x + 2log6 (3x) − log6 (2x3 ) → log6 (x3 + 3x2 ) − log6 (2x3 ) → log6 ( x 2x
3 ) → log6 ( 2x )
4) Reversing the rule used in Q 2 gives: logx ( yz ) = logx y − logx z
7
3
log2 ( 5x
) → log2 ( 5x3 ) → (reducing the fraction first) log2 5x3 − log2 3
3x4
Practice
Expand each logarithmic expression:
1. log5 (ab)
2. log6 √a
3b
3. log6 ab
c
2
4. If v = logx ( 4zy3 ) expand v
3
4x
5. log2 ( √
)
y
6. If R = log3 ( 2GM
) expand R
c2
Condense each logarithmic expression:
7. log5 A + log5C
8. 12 log2C − log2 B
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9.
10.
11.
12.
13.
2logb x + 2logb y
6log10 a + log10 b
2log3 a + 4log3 b − log3 c
1
2 log4 w − 5log4 z
(log10 x + log10 y) − log10 w
Simplify:
2
1
5
14. log10 A3 − log10 B 3 + log10 A 3 + log10 B 3
2
9 A −2log9 B
15. log
log9 A2 +log9 B3
16. 2ln(AB) − ln( BA )
4