First verify that you are using a common logarithm table. In a common logarithm table, the values for 1.0 through 9.99 will range from 0 to 1. (In a natural logarithm table, the values for 1.0 through 9.99 will range from 0 to approximately 2.3. The illustration above is of a table of natural logarithms.)
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Express the two numbers in scientific notation, i.e., of the form A × 10B where A is greater than or equal to 1 and less than 10.
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For each number, look up the "A" value in the table of logarithms.
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Add the two values you looked up. If the result is greater than one, then subtract one.
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Look in the body of the table for the number closest to the result and use the row and column headers to determine what number it corresponds to. This is the value of "A" for the product.
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Add the two "B" values, plus 1 more if you subtracted 1 in step 3. This is the value of "B" for the product.
Example: Compute 25.2 × 120.
| N | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| 1.2 | 07918 | 08279 | 08636 | 08991 | 09342 | 09691 | 10037 | 10380 | 10721 | 11059 |
| ... |
| 2.5 | 39794 | 39967 | 40140 | 40312 | 40483 | 40654 | 40824 | 40993 | 41162 | 41330 |
| ... |
| 3.0 | 47712 | 47857 | 48001 | 48144 | 48287 | 48430 | 48572 | 48714 | 48855 | 48996 |
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Rewrite 25.2 = 2.52 × 101 and 120 = 1.20 × 102.
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According to the table, log(2.52) = .40140 and log(1.20) = .07918.
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Compute .40140 + .07918 = .48058. The result is less than 1 so no adjustment is necessary.
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The value .48058 lies between .48001 and .48144, but it's closer to .48001, which corresponds to the value 3.02.
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Adding the exponents 1 + 2 = 3. We did not have an extra 1 from step 3, so the total exponent is 3.
The product is therefore approximately 3.02 × 103 = 3020. (The exact answer is 3024.)
Practical example: The volume of a sphere is given by the formula V = 4πr3/3. What is the approximate volume of a sphere whose radius is 30.8 meters?
Since the measurement is accurate to only three significant digits, we can perform our intermediate computations to only three significant digits. Since π is approximately 3, the value we expect is approximately 4 × 3 × 303 / 3 = 4 × 30^3 = 4 × 27,000 ≈ 100,000.
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Using the rule of logarithms, log(V) = log(4) + log(π) + 3 log(r) - log(3).
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Use the rule of logarithms again to convert log(4) - log(3) = log(4/3) = log(1.33).
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From the table, log(3.08) = .489, log(1.33) = .124.
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Most log tables also provide the logarithms of well-known constants, so you can look up log(π) = .497.
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Calculate 3 log(r) = 3 × .489 = 1.467. This calculation is simple enough that it can be done in your head.
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Add the pieces together. log(1.33) + log(π) + 3 log(r) = .124 + .497 + 1.467 = 2.088.
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Look up .088 in the logarithm table: It is closest to log(1.22) = .08636.
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Therefore, the volume of the sphere is approximately 122,000 cubic meters. (The correct answer is 122,388.541...)
We have arrived at the correct answer (to three significant digits) using only addition and multiplication by 3. Computing this result by hand without the use of logarithms would have been significantly more difficult.
Shortcuts
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Step 1 can be shortened to "move the decimal point to immediately after the first nonzero digit."
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Replace step 5 with "Move the decimal place to the location that makes sense based on the values being multiplied."
Example: Compute 25.2 × 120.
-
Rewrite 25.2 as 2.52 and 120 as 1.20.
-
According to the table, log(2.52) = .40140 and log(1.20) = .07918.
-
Compute .40140 + .07918 = .48058. The result is less than 1 so no adjustment is necessary.
-
The value .48058 lies between .48001 and .48144, but it's closer to .48001, which corresponds to the value 3.02.
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Make a rough estimate of the expected answer. 25.2 × 120 will be higher than 25 × 100 = 2500 but smaller than 30 × 200 = 6000. Starting with the value 3.02, moving the decimal three places to the right results in a value in this range: 3020. (The exact answer is 3024.)
E-Store
First verify that you are using a common logarithm table. In a common logarithm table, the values for 1.0 through 9.99 will range from 0 to 1. (In a natural logarithm table, the values for 1.0 through 9.99 will range from 0 to approximately 2.3. The illustration above is of a table of natural logarithms.)
Express the two numbers in scientific notation, i.e., of the form A × 10B where A is greater than or equal to 1 and less than 10.
For each number, look up the "A" value in the table of logarithms.
Add the two values you looked up. If the result is greater than one, then subtract one.
Look in the body of the table for the number closest to the result and use the row and column headers to determine what number it corresponds to. This is the value of "A" for the product.
Add the two "B" values, plus 1 more if you subtracted 1 in step 3. This is the value of "B" for the product.
Example: Compute 25.2 × 120.
Rewrite 25.2 = 2.52 × 101 and 120 = 1.20 × 102.
According to the table, log(2.52) = .40140 and log(1.20) = .07918.
Compute .40140 + .07918 = .48058. The result is less than 1 so no adjustment is necessary.
The value .48058 lies between .48001 and .48144, but it's closer to .48001, which corresponds to the value 3.02.
Adding the exponents 1 + 2 = 3. We did not have an extra 1 from step 3, so the total exponent is 3.
The product is therefore approximately 3.02 × 103 = 3020. (The exact answer is 3024.)
Practical example: The volume of a sphere is given by the formula V = 4πr3/3. What is the approximate volume of a sphere whose radius is 30.8 meters?
Since the measurement is accurate to only three significant digits, we can perform our intermediate computations to only three significant digits. Since π is approximately 3, the value we expect is approximately 4 × 3 × 303 / 3 = 4 × 30^3 = 4 × 27,000 ≈ 100,000.
Using the rule of logarithms, log(V) = log(4) + log(π) + 3 log(r) - log(3).
Use the rule of logarithms again to convert log(4) - log(3) = log(4/3) = log(1.33).
From the table, log(3.08) = .489, log(1.33) = .124.
Most log tables also provide the logarithms of well-known constants, so you can look up log(π) = .497.
Calculate 3 log(r) = 3 × .489 = 1.467. This calculation is simple enough that it can be done in your head.
Add the pieces together. log(1.33) + log(π) + 3 log(r) = .124 + .497 + 1.467 = 2.088.
Look up .088 in the logarithm table: It is closest to log(1.22) = .08636.
Therefore, the volume of the sphere is approximately 122,000 cubic meters. (The correct answer is 122,388.541...)
We have arrived at the correct answer (to three significant digits) using only addition and multiplication by 3. Computing this result by hand without the use of logarithms would have been significantly more difficult.
Shortcuts
Step 1 can be shortened to "move the decimal point to immediately after the first nonzero digit."
Replace step 5 with "Move the decimal place to the location that makes sense based on the values being multiplied."
Example: Compute 25.2 × 120.
Rewrite 25.2 as 2.52 and 120 as 1.20.
According to the table, log(2.52) = .40140 and log(1.20) = .07918.
Compute .40140 + .07918 = .48058. The result is less than 1 so no adjustment is necessary.
The value .48058 lies between .48001 and .48144, but it's closer to .48001, which corresponds to the value 3.02.
Make a rough estimate of the expected answer. 25.2 × 120 will be higher than 25 × 100 = 2500 but smaller than 30 × 200 = 6000. Starting with the value 3.02, moving the decimal three places to the right results in a value in this range: 3020. (The exact answer is 3024.)