For the following, assume that x, y, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1.
Definitions
1. log_{a} x = N means that a^{N} = x.
2. log x means log_{10} x. All log_{a} rules apply for log. When a logarithm is written without a base it means common logarithm.
3. ln x means log_{e} x, where e is about 2.718. All log_{a} rules apply for ln. When a logarithm is written "ln" it means natural logarithm.
Note: ln x is sometimes written Ln x or LN x.
Rules
1. Inverse properties: log_{a} a^{x} = x and a^{(log}a ^{x)} = x
2. Product: log_{a} (xy) = log_{a} x + log_{a} y
3. Quotient:
4. Power: log_{a} (x^{p}) = p log_{a} x
5. Change of base formula:
Careful!!
log_{a} (x + y) ≠ log_{a} x + log_{a} y
log_{a} (x – y) ≠ log_{a} x – log_{a} y
