RULES OF LOGARITHMS

For the following, assume that x, y, a, and b are all positive. Also assume that a ≠ 1, b ≠ 1. Definitions 1. loga x = N means that aN = x. 2. log x means log10 x. All loga rules apply for log. When a logarithm is written without a base it means common logarithm. 3. ln x means loge x, where e is about 2.718. All loga 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:   loga ax = x   and   a(loga x) = x 2. Product:  loga (xy) = loga x + loga y 3. Quotient:   4. Power:   loga (xp) = p loga x 5. Change of base formula:  Careful!! loga (x + y) ≠ loga x + loga y loga (x – y) ≠ loga x – loga y

Since logarithms are nothing more than exponents, these rules come from the rules of exponents. Let a be greater than 0 and not equal to 1, and let n and m be real numbers.

Exponential Rule 1:  

Example: Let a = 5, n = 2, and m = 6.  and  

Exponential Rule 2:  

Example: Let a = 5, n = 2, and m = 6.  and  

Exponential Rule 3:  

Example: Let a = 5, n = 2, and m = 6.  and