Before computers and calculators, logarithms were quickly calculated using logarithmic tables. Once you figure out how they are used, these tables can help you quickly calculate logarithms or multiply large numbers.
Steps
Method 1 of 3: Read a table of logarithms
Step 1. Understand what a logarithm is
102 is 100. 103 is 1000. The powers of 2 and 3 are logarithms to base 10, or common logarithms, of 100 and 1000. In general ab = c can be rewritten as logtoc = b. Therefore, saying "10 to the second power is 100" is equivalent to saying "the base 10 logarithm of 100 is 2." Logarithm tables are base 10 (they use the common logarithm) so a must always be 10.
- Multiply two numbers by adding their powers. For example: 102 * 103 = 105, or 100 * 1000 = 100,000.
- The natural logarithm, which is represented as "ln", is the logarithm to base e, where "e" is the constant 2,718. This is a useful number in many areas of mathematics and physics. You can use tables of natural logarithms in the same way that you use common or base 10 logarithmic tables.
Step 2. Identify the characteristic of the number whose natural logarithm you want to find
15 is between 10 (101) and 100 (102) so its logarithm will be between 1 and 2, that is, it will be 1, something. 150 is between 100 (102) and 1000 (103), so your logarithm will be between 2 and 3, or it will be 2, something. That "something" is known as a mantissa. This is what you will find in the logarithmic table. What comes before the decimal point (1 in the first case, 2 in the second) is the characteristic.
Step 3. Slide your finger down to the corresponding row in the table using the far left column
This column will show you the first two digits, or the first three in larger tables, of the number for which you want to find the logarithm. If you are going to find the log of 15, 27 in a normal logarithmic table, go to the row marked 15. If you are going to find the log of 2, 57 go to the row marked 25.
- Sometimes the numbers in this list have a decimal point, so look for 2, 5 instead of 25. You can ignore this decimal point as it will not affect your result.
- Also ignore the decimal points in the number whose logarithm you want to find, since the mantissa for the logarithm of 1,527 is no different than the mantissa for the logarithm of 152,7.
Step 4. In the corresponding row, slide your finger over the corresponding column
This column is the one marked with the next digit of the number whose logarithm you want to find. For example, if you want to find the logarithm of 15, 27 your finger should be in the row marked 15. Slide your finger along the row to the right to find column 2. You will be pointing at the number 1818. Write it down.
Step 5. If your log table has a mean difference table, slide your finger over the column in the table marked with the next digit of the number you are looking for
For 15, 27 this number is 7. Your finger will currently be in row 15 and column 2. Slide it to row 15 and in the difference of means go to column 7. Now your finger will point to number 20. Write it down.
Step 6. Add the numbers you found in the last two steps
For 15, 27 you will get 1838. This is the mantissa of the logarithm of 15, 27.
Step 7. Add the characteristic to them
Because 15 is between 10 and 100 (101 and 102), the logarithm of 15 must be between 1 and 2 (1, something) so the characteristic will be 1. Combine the characteristic with the mantissa to get the final answer. Finally you will see that the logarithm of 15.27 is 1.1838.
Method 2 of 3: Find the antilogarithm
Step 1. Understand the use of the antilogarithm table
Use it when you have the logarithm of a number and want to find the number in question. In formula 10 = x, "n" is the common logarithm or base 10 of "x". If you have the "x", find the "n" using the logarithm table. If you have the "n", find the "x" using the antilogarithm table.
The antilogarithm is also known as the inverse logarithm
Step 2. Write the characteristic
It is the number that comes before the decimal point. If you are looking for the antilogarithm of 2, 8699 the characteristic is 2. Mentally remove it from the number you are looking for, but be sure to write it down so you don't forget it (you'll need it later).
Step 3. Find the row that matches the first part of the mantissa
In the case of 2, 8699 the mantissa is 0, 8699. Most tables of antilogarithms, like most tables of logarithms, have two digits in the leftmost column so run your finger down that column until you find 0, 86.
Step 4. Slide your finger over the column that is marked with the next number of the mantissa
For 2, 8699 slide your finger along the row marked 0.86 to find the intersection with column 9. This box should read 7396. Write down this number.
Step 5. If your antilogarithm table has a mean difference table, slide your finger over the column in the table marked with the next digit of the number you are looking for
Make sure to keep your finger in the same row. In this case you should slide your finger over the last column of the table, column 9. The intersection of row 0, 86 and column 9 of the mean difference is 15. Write down that number.
Step 6. Add the two numbers from the previous steps
For this example, the numbers are 7396 and 15. Add them together to get 7411.
Step 7. Use the characteristic to locate the decimal point
The characteristic in the example was 2. This means that the answer is between 102 and 103 or between 100 and 1000. For the number 7411 to fall between 100 and 1000, the decimal point must come after the three digits, so the number will be approximately 700 instead of 70, which is too small or 7000, which is too large. Therefore the final answer is 741, 1.
Method 3 of 3: Multiply Numbers Using Tables of Logarithms
Step 1. Understand how to multiply numbers using their logarithms
You already know that 10 * 100 = 1000. Expressed in terms of powers (or logarithms), 101 * 102 = 103. Overall 10x * 10and = 10x + y. So the sum of the logarithms of two different numbers is the logarithm of the product of those numbers. You can multiply two numbers of the same base by adding their powers.
Step 2. Find the logarithms of the two numbers you want to multiply
Use the above method to find the logarithms. For example, if you want to multiply 15, 27 by 48, 54, you would first find the log of 15, 27 which is 1.1838 and the log of 48.54 which is 1.6861.
Step 3. Add the two logarithms to find the logarithm of the solution
In this example, add 1.1838 and 1.6861 to get 2.8699. This number is the logarithm of the answer.
Step 4. Find the antilogarithm of the result from the previous step to find the solution
You can do this by looking for the number in the body of the table that is closest to the mantissa of that number (8699). However, the most efficient and reliable method is to find the answer in the antilogarithm table, as explained in the previous method. For this example you will get 741, 1 as a result.
Advice
- Always do the calculations on a sheet of paper and not mentally, as they are complicated numbers and it can get difficult.
- Read the header of the page carefully. A book of logarithms is about 30 pages long and if you look at the wrong page you will get an incorrect answer.
Warnings
- Make sure the readings come from the same row. Rows and columns are sometimes jumbled up due to the small font size and spacing.
- Most tables have a precision of three or four digits. If you look up the antilogarithm of 2.8699 using a calculator, the answer will be rounded to 741.2, but the answer you would get using the table of logarithms is 741.1. This is because tables round numbers. If you need a more precise answer, use a calculator or another method other than tables of logarithms.
- Use the methods explained in this article for base 10 or common logarithm tables and make sure the numbers you are looking for are in base 10 or scientific notation format.