Find or estimate the value of each variable mentally.
9.2
Activity
A Table of Numbers
2
0.3010
3
0.4771
4
0.6021
5
0.6990
6
0.7782
7
0.8451
8
0.9031
9
0.9542
10
1
20
1.3010
30
1.4771
40
1.6021
50
1.6990
60
1.7782
70
1.8451
80
1.9031
90
1.9542
100
2
200
2.3010
300
2.4771
400
2.6021
500
2.6990
600
2.7782
700
2.8451
800
2.9031
900
2.9542
1,000
3
2,000
3.3010
3,000
3.4771
4,000
3.6021
5,000
3.6990
6,000
3.7782
7,000
3.8451
8,000
3.9031
9,000
3.9542
10,000
4
Analyze the table and discuss with a partner what you think the table tells us.
Use the table to find the value of the unknown exponent that makes each equation true.
Notice that some values in the columns labeled are whole numbers, but most are decimals. Why do you think that is?
9.3
Activity
Hello, Logarithm!
Here are two true equations based on the information from the table:
What values could replace the “?” in these equations to make them true?
Estimate the value of ? Be prepared to explain how you know.
The term log is short for logarithm. Discuss the following questions with a partner, and record your responses:
What do you think logarithm means or does?
Next to “log” is a subscript—a number or letter printed smaller and below the line of text—that is called the “base.” What do you think the base of the logarithm tells us?
What about the other two numbers on either side of the equal sign (for example, the 100 and the 2 in )? What do they tell us?
Student Lesson Summary
We know how to solve equations such as or by thinking about integer powers of 10. The solutions are and . What about an equation such as ?
Because and , we know that is between 2 and 3. We can use a logarithm to represent the exact solution to this equation and write it as:
The expression is read “the log, base 10, of 250.”
The small, slightly lowered “10” refers to the base of 10.
The 250 is the value of the power of 10.
is the value of the exponent that makes equal 250.
In the specific case where the base of the logarithm is 10, the “log” can be written without the number 10. For example, can also be written as , and this expression is read “the log of 250.”
One way to estimate logarithms is with a logarithm table. For example, using this base 10 logarithm table we can see that is between 2.3010 and 2.4771.
2
0.3010
3
0.4771
4
0.6021
5
0.6990
6
0.7782
7
0.8451
8
0.9031
9
0.9542
10
1
20
1.3010
30
1.4771
40
1.6021
50
1.6990
60
1.7782
70
1.8451
80
1.9031
90
1.9542
100
2
200
2.3010
300
2.4771
400
2.6021
500
2.6990
600
2.7782
700
2.8451
800
2.9031
900
2.9542
1,000
3
2,000
3.3010
3,000
3.4771
4,000
3.6021
5,000
3.6990
6,000
3.7782
7,000
3.8451
8,000
3.9031
9,000
3.9542
10,000
4
Glossary
logarithm
The logarithm to base 10 of a number , written , is the exponent you raise 10 to get , so it is the number that makes the equation true. Logarithms to other bases are defined the same way with 10 replaced by the base, e.g. is the number that makes the equation true. The logarithm to the base is called the natural logarithm, and is written .
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