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The expression can be read as: “The log, base 10, of 1,000 is 3.”
It can be interpreted as: “The exponent to which we raise a base 10 to get 1,000 is 3.”
Take turns with a partner reading each equation out loud. Then, interpret what they mean.
| 1 | 0 |
| 2 | 1 |
| 3 | 1.5850 |
| 4 | 2 |
| 5 | 2.3219 |
| 6 | 2.5850 |
| 7 | 2.8074 |
| 8 | 3 |
| 9 | 3.1699 |
| 10 | 3.3219 |
| 11 | 3.4594 |
| 12 | 3.5845 |
| 13 | 3.7004 |
| 14 | 3.8074 |
| 15 | 3.9069 |
| 16 | 4 |
| 17 | 4.0875 |
| 18 | 4.1699 |
| 19 | 4.2479 |
| 20 | 4.3219 |
| 21 | 4.3923 |
| 22 | 4.4594 |
| 23 | 4.5236 |
| 24 | 4.5850 |
| 25 | 4.6439 |
| 26 | 4.7004 |
| 27 | 4.7549 |
| 28 | 4.8074 |
| 29 | 4.8580 |
| 30 | 4.9069 |
| 31 | 4.9542 |
| 32 | 5 |
| 33 | 5.0444 |
| 34 | 5.0875 |
| 35 | 5.1293 |
| 36 | 5.1699 |
| 37 | 5.2095 |
| 38 | 5.2479 |
| 39 | 5.2854 |
| 40 | 5.3219 |
These equations express the same relationship between 2, 16, and 4:
| exponential form | logarithmic form | |
|---|---|---|
| a. | ||
| b. | ||
| c. | ||
| d. | ||
| e. | ||
| f. | ||
| g. | ||
| h. | ||
| i. | ||
| j. |
Many relationships that can be expressed with an exponent can also be expressed with a logarithm. Let’s look at this equation: The base is 2 and the exponent is 7, so it can be expressed as a logarithm with base 2:
In general, an exponential equation and a logarithmic equation are related as shown here:
Exponents can be negative, so a logarithm can have negative values. For example, , which means that .
An exponential equation cannot always be solved by observation. For example, does not have an obvious solution. The logarithm gives us a way to represent the solution to this equation: . The expression is approximately 4.2479, but is an exact solution.