4.3 Laws of Logarithms

2 Laws of Logarithms  Just like the rules for exponents there are corresponding rules for logs that allow you to rewrite the log of a product, the log of a quotient, or the log of a power.

3 Log of a Product  Logs are just exponents  The log of a product is the sum of the logs of the factors: log b xy = log b x + log b y Ex: log (25 ·125) = log 25 + log 125

4 Log of a Quotient  Logs are exponents  The log of a quotient is the difference of the logs of the factors: log b = log b x – log b y Ex: ln ( ) = ln 125 – ln 25

5 Log of a Power  Logs are exponents  The log of a power is the product of the exponent and the log: log b x n = n∙log b x Ex: log 3 2 = 2 ∙ log 3

6 Rules for Logarithms  These same laws can be used to turn an expression into a single log: log b x + log b y = log b xy log b x – log b y = log b n∙log b x = log b x n

7 log b (xy) = log b x + log b y Express as a sum and difference of logarithms: = log 3 A + log 3 B Examples log b ( ) = log b x – log b ylog b x n = n log b x _______________________________ Solve: x = log 3 30 – log 3 10 = log 3 3 Evaluate: = 2 = – log 3 C x = 1 = log 3 AB

8 Sample Problem  Express as a single logarithm: 3log 7 x + log 7 (x+1) - 2log 7 (x+2)  3log 7 x = log 7 x 3  2log 7 (x+2) = log 7 (x+2) 2 log 7 x 3 + log 7 (x+1) - log 7 (x+2) 2 log 7 (x 3 ·(x+1)) - log 7 (x+2) 2  log 7 (x 3 ·(x+1)) - log 7 (x+2) 2 =

To use a calculator to evaluate logarithms with other bases, you can change the base to 10 or “e” by using either of the following: For all positive numbers a, b, and x, where a ≠ 1 and b ≠ 1: Example: Evaluate log 4 22 ≈ Change of Base Formula