Algebra II

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

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. 

   Note: Don't confuse with .

   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. .

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

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. 

3. log ₃ 25

4. 

5. log ₃ 1.5

6. log ₃ 200

Example 3

Rewrite each expression as the logarithm of a single quantity.