9.5 Properties of Logarithms

1. 9.5 Properties 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: • logb xy = logb x + logb y • Ex: log(25 ·125) = log25 + log125

4. Log of a Quotient • Logs are exponents • The log of a quotient is the difference of the logs of the factors: • logb = logb x – logb y • Ex: ln( ) = ln125 – ln25

5. Log of a Power • Logs are exponents • The log of a power is the product of the exponent and the log: • logb xn = n∙logb x • Ex: log 32 = 2 ∙ log 3

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

7. Examples logb(xy) = logb x + logb y logb( ) = logb x – logb y logb xn = n logb x _______________________________ as a sum and difference of logarithms: Express = log3A + log3B = log3 AB – log3C Evaluate: Solve: x = log330 – log310 = log33 = 2 = x = 1

8. Sample Problem • Express as a single logarithm: 3log7x + log7(x+1) - 2log7(x+2) • 3log7x = log7x3 • 2log7(x+2) = log7(x+2)2 log7x3 + log7(x+1) - log7(x+2)2 log7(x3·(x+1)) - log7(x+2)2 • log7(x3·(x+1)) - log7(x+2)2 =

9. Change of Base Formula For all positive numbers a, b, and x, where a ≠ 1 and b ≠ 1: 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: Example: Evaluate log4 22 ≈ 2.2295