1. © 2008 Shirley Radai Properties of Logarithms

2. What is a logarithm?  Logarithms are really powers (exponents). The Relationship: "log b (x) = y" means the same thing as "b y = x".  Since 2 3 =8, 3 is called the logarithm of 8 with base 2. We write 3=log 2 8. © 2008 Shirley Radai

3. Expanding Logarithmic Expressions  To “expand” a logarithmic expression means to take a log with multiple factors inside it and rewrite it into different logs with single factors inside. Multiplying inside a log turns into addition outside the log if the bases are the same. Division inside a log turns into subtraction outside the log if the bases are the same. An exponent inside a log is moved to the front of the log to become a multiplier if the bases are the same. © 2008 Shirley Radai

4. Examples Since we have multiplication inside the log (5x), it becomes addition. Since we have division inside the log (16/x), it becomes subtraction. © 2008 Shirley Radai

5. Examples (cont’d) Since there is an exponent inside x 4, the exponent goes out front of the log. © 2008 Shirley Radai

6. Properties of Logarithms  log b (b) = 1, for any base b, because b 1 = b.  log b (1) = 0, for any base b, because b 0 = 1.  log b (a) is undefined if a is negative.  log b (0) is undefined for any base b.  log b (b n ) = n, for any base b. © 2008 Shirley Radai