Algebra II

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. 
7. Note: Don't confuse with . 
8. To find the latter, first evaluate each log separately and then do the division.
9. log _bM ^x= x log _bM 
10. If log _bx = log _by , then x = y. 
11. . 

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
5. 
6. 
7. 
8. 

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 
7. 
8. 
9. 
10. 
11. 
12. 

Example 3

Rewrite each expression as the logarithm of a single quantity.