How to Divide Using Logarithms. A logarithm is nothing more than an exponent; it's just expressed in a different manner. Instead of saying that 2 raised to the 3rd power (exponent 3) is 8, say that log 2 of 8 is 3. In other words, 2 raised to what power gives 8? Dividing using logarithms is as easy as dividing using exponents.
Choose two numbers that can't be easily divided using pencil and paper. For example, 82,310 can't be easily divided by 162.
Express the numbers in terms of base 10 logarithms. The number 82,310 can be expressed as log82310 (the base of 10 is understood) and 162 can be expressed as log162.
Use a logarithm table to determine the logarithms of both expressions. For example, log82310 is 4.9153998. To do this, look up log8.231 to get the numbers to the right of the decimal point, then add a 4 to the left of the decimal. Log162 is 2.2095150
Subtract 2.21 from 4.915 to get 2.7058637.
Use the logarithm table to find the antilog of 2.7058637. To do this, look up .7058637, then move the decimal place of the result to the right two places. The answer is 508.
TL;DR (Too Long; Didn't Read)
Before the existence of calculators, logarithms and logarithm tables saved scientists many hours of "number crunching." Logarithms still have uses today.
You will not get a correct answer by subtracting log162 from log82310 to get log82238. You must find the logs, subtract them, and then find the antilogs of the result.