Properties of Logarithms

The properties of logarithms assume the following about the variables M, N, b, and x.

equation

  1. log bb = 1

  2. log b 1 = 0

  3. log bb x = x

  4. b logbx = x

  5. log b ( MN) = log b ( M) + log b ( N)

  6. equation

    Note: Don't confuse equation with equation.

    To find the latter, first evaluate each log separately and then do the division.

  7. log bM x = x log bM

  8. If log bx = log by , then x = y.

  9. equation.

This is known as the change of base formula.

Example 1

Simplify each of the following expressions.

  1. log 7 7

  2. log 5 1

  3. log 44 3

  4. 6 log65

  1. equation

  2. equation

  3. equation

  4. equation

Example 2

If log 3 5 ≈ 1.5, log 3 3 = 1, and log 3 2 ≈ 0.6, approximate the following by using the properties of logarithms.

  1. log 3 10

  2. equation

  3. log 3 25

  4. equation

  5. log 3 1.5

  6. log 3 200

  1. equation

  2. equation

  3. equation

  4. equation

  5. equation

  6. equation

Example 3

Rewrite each expression as the logarithm of a single quantity.

  1. equation

  2. equation

  1. equation

  2. equation