What are the steps for condensing logarithms?

Logarithms are basically inverse exponential expressions. For that reason, many of the rules for combining or condensing exponents can be applied in reverse to logarithms. Simplifying and condensing logarithms should be done without a calculator.

1. Determine what terms can be condensed

   Coefficients multiplied by logarithms can be converted to exponents within the logarithm. For example, 3*log4(2) = log4(2^3) = log4(8). Logarithms with the same base can be condensed if they are added or subtracted from one another. The subscript number after the word "log" is the base that must be raised to an unknown power to generate the number in the parentheses, also called the argument. In the expression log4(2), 4 is the base and 2 is the argument. log4(2) and log4(8) can be condensed, but log2(4) and log8(4) cannot.

2. Add or subtract logarithms with like bases

   If the logarithms are added, multiply the numbers inside the parenthesis. If they are subtracted, divide the numbers inside the parentheses. For example, log4(2) + log4(8) = log4(16).

3. Solve or simplify logarithms that are easily solvable

   The answer to a logarithm is the power that would produce the argument when applied to the base. Keep some basic logarithm rules in mind. If the argument of any logarithm is 1, the answer is 0. If the base and argument are equal, the answer is 1. For a condensed logarithm like log4(16), use your knowledge of squares. 4^2 = 16, so log4(16) = 2.