Unit 4, Lesson 9
What Is a Logarithm?
Integrated Math 3
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Find or estimate the value of each variable mentally.
| 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 |
What values could replace the “?” in these equations to make them true?
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.”
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 |
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 .