Properties of Logarithms

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

  1. log bb = 1 
  2. log b1 = 0 
  3. log bb x= x
  4. b logbx= x
  5. log b( MN) = log b( M) + log b( N
  6. Note: Don't confuse with
  7. To find the latter, first evaluate each log separately and then do the division.
  8. log bM x= x log b
  9. If log bx = log by , then x = y

This is known as the change of base formula.

Example 1

Simplify each of the following expressions.

  1. log 7
  2. log 5
  3. log 44 3
  4. 6 log65

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. log 3 25 
  3. log 3 1.5 
  4. log 3 200 

Example 3

Rewrite each expression as the logarithm of a single quantity.

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