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3.3 Logarithms
1
In this section we will introduce a new concept which is the logarithm. First we will give the definition of a logarithm.
Definition of Logarithm: For all positive numbers b, where b 1, and all positive numbers r, t
bb r is equivalent to log r=t.
This key statement should be memorized. The abbreviation log is used for logarithm. logbr is read as “the logarithm of r to the base b.
Meaning of logbr: A logarithm is an exponent; logbr is the exponent on the base b that yields the number t.
We can use the definition of logarithm to write exponential statements in logarithmic form and logarithmic statements in exponential form.
To remember the location of the base and exponent in each form, refer to the following diagram:
texponential form: b r
base
exponent
blogaritmic form: log r=t
base
exponent
Note: r is often called the result
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Step1. Write the equation in exponential form by using the definition of a logarithm.
logbr = t is equivalent to bt = rStep 2. Use Property 1 of this chapter solve the equation. If b > 0 and b 1, and m and n are real numbers, then bn = bm if and only if n = m.
1. Write the equation in exponential form.
2. Solve by matching the bases by rewriting 16 using a base of 2.
x2 16
Example 1. Solve: log216 = x
Solution:x 42 2
The solution set is {4}.
x 4
Logarithmic Equations
A logarithmic equation is an equation with a logarithm in at least one term. Our objective here is show how changing from logarithmic form to exponential form will enable us to solve other types of equations.
Procedure: To solve equations of the form logbr = t.
Solve: log6216 = x
Your Turn Problem #1
Answer: {3}
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2. Write the equation in exponential form.
3. Solve by matching the bases. Rewrite 64 by using a base of 2.
x2 64
Example 2. Evaluate: log264
Solution:
x 62 2
1. Set the logarithm equal to x.
2log 64 x
x 6Answer:
Solve: log100.0001
Your Turn Problem #2
Answer: 4
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x5 5
1x 25 5
Solution:
5 Evaluate: Examp lole 3 g. 5
2. Write the equation in exponential form.3. Solve by matching the bases.
1. Set the logarithm equal to x.
5log 5 = x
1x
2Answer:
Your Turn Problem #3
43Evaluate: log 3
1Answer:
4
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1. Write the equation in exponential form.
2. Solve by matching the bases.
x 15
25
x 25 5
The solution set is {-2}.
x 2
Solution:
5
1. Solve: Ex log = x
2m e 4
5a pl
Your Turn Problem #4
3
1Solve: log x
81
Answer: 4
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1. Write the equation in exponential form.
2. Evaluate the left hand side of the equation by using a calculator or writing the expression in radical form.
2327 x
The solution set is {9}.
x 9
Solution:
27
2 Solve: logExam x ple 5 = .
3
23 27 x
23 x
Your Turn Problem #5
8
4Solve: log x
3
Answer: 16
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1. Write the equation in exponential form.
2. Get x by itself by squaring both sides.
12x 4
The solution set is {16}.
x 16
Solution:
x
1 Solve: logExam 4 pl 6. = e
2
21
22x 4
Your Turn Problem #6
x
1Solve: log 5
2
Answer: 25
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x17 1
The solution set is {0}.
x 0
Properties of logarithms
For any positive real number b, we know that b1 = b, and b0 = 1. Writing these two statements in logarithmic form gives the following two properties of logarithms.
For any positive real number b, b 1, logbb = 1 and logb1 = 0.
Solution:
17 Solve: logExample 7. 1 = x
This equation is solved by using the property from the previous slide. It can also be solved by writing the equation in exponential form.
Your Turn Problem #7
7Solve: log 7 x
Answer: 1
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Three more Properties of Logarithms
Product Rule for Logarithms
b b bFor any positive numbers b, r, and s, where b 1, log rs log r log s.
Quotient Rule for Logarithms
b b b
rFor any positive numbers b, r, and s, where b 1, log log r log s.
s
Power Rule for Logarithms
pb b
If r is any positive real number, b is a positive real number other than 1 and p is any real
number, then log r plog r.
Using these properties enables us to solve other types of logarithmic equations. For the following examples, use the product or quotient rule to write the logarithms as a single logarithm. Then change from logarithmic form to exponential form to solve the following equations.
Next Slide
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1. Use the product rule to rewrite as a single logarithms.
2. Change from logarithmic form to exponential form.
5log 3x 2
25 3x
Solution:
5 5 log xExample 8. log 3 2
3x 25
25x
3
Answer:
253
Your Turn Problem #8
3 3Solve: log 4 log x 2
4Answer:
9
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1. Use the product rule to rewrite as a single logarithms.
2. Change from logarithmic form to exponential form and solve.
2log x(x 7) 3 32 x(x 7)
Solution:
2 2 log xExample log 9. (x 7) 3
2x 7x 8 2x 7x 8 0
(x 8)(x 1) 0
x 8 or x 1
Recall from the definition on slide 1, b and r must be positive. Therefore –1 must be rejected as a solution, since it leads to the logarithm of a negative number in the original equation:log2(-1)+log2(-1-7)=2 Therefore, the only solution is 8.
Answer: 8
Your Turn Problem #9
6 6Solve: log (x 1) log (x 4) 2
Answer: 8
The End.B.R.1-11-07
