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Understand what a logarithm is. Identify the characteristic of the number whose log you want to find. Slide your finger down to the appropriate row on the table using the leftmost column. On the appropriate row, slide your finger over to the appropriate column. If your log table has a mean difference table, slide your finger over to the column in that table marked with the next digit of the number you're looking up. Add the numbers found in the two preceding steps together. Add the characteristic.
102 is 100. 103 is 1000. The powers 2 and 3 are the base-10 logarithms of 100 and 1000. In general, ab = c can be rewritten as logac = b. So, saying "ten to the power of two is 100" is equivalent to saying "the base-ten log of 100 is two." Each logarithmic table is only usable with a certain base (a in the equation above). By far the most common type of log table uses base-10 logs, also called the common logarithm. Multiply two numbers by adding their powers. For example: 102 * 103 = 105, or 100 * 1000 = 100,000. The natural log, represented by "ln", is the base-e log, where e is the constant 2.718. This is a useful number in many areas of math and physics. You can use natural log tables in the same way that you use common, or base-10, log tables. Let's say you want to find the base-10 log of 15 on a common log table. 15 lies between 10 (101) and 100 (102), so its logarithm will lie between 1 and 2, or be 1.something. 150 lies between 100 (102) and 1000 (103), so its logarithm will lie between 2 and 3, or be 2.something. The .something is called the mantissa; this is what you will find in the log table. What comes before the decimal point (1 in the first example, 2 in the second) is the characteristic. This column will show the first two or, for some large log tables, three digits of the number whose logarithm you're looking up. If you're looking up the log of 15.27 in a normal log table, go to the row marked 15. If you're looking up the log of 2.57, go to the row marked 25. Sometimes the numbers in this row will have a decimal point, so you'll look up 2.5 rather than 25. You can ignore this decimal point, as it won't affect your answer. Also ignore any decimal points in the number whose logarithm you're looking up, as the mantissa for the log of 1.527 is no different from that of the log of 152.7. This column will be the one marked with the next digit of the number whose logarithm you're looking up. For example, if you want to find the log of 15.27, your finger will be on the row marked 15. Slide your finger along that row to the right to find column 2. You will be pointing at the number 1818. Write this down. For 15.27, this number is 7. Your finger is currently on row 15 and column 2. Slide it over to row 15 and mean differences column 7. You will be pointing at the number 20. Write this down. For 15.27, you will get 1838. This is the mantissa of the logarithm of 15.27. Since 15 is between 10 and 100 (101 and 102), the log of 15 must be between 1 and 2, so 1.something, so the characteristic is 1. Combine the characteristic with the mantissa to get your final answer. Find that the log of 15.27 is 1.1838.