1. Unit 11: Logarithms, day 3 3 properties to Expand and Condense Logarithmic Expressions

2. The Product Property Definition: The log of a product can be expanded into the SUM of the logs of the factors log b mn = log b m + log b n (EXPANDING) EX: log 3 7x = log 3 7 + log 3 x EX: log 2 15 = log 2 3 + log 2 5 (since 3*5 = 15)

3. The Product Property Definition: The SUM of logs with the same base can be condensed into the log of the product log b m + log b n = log b mn (CONDENSING) EX: log 3 7 + log 3 x = log 3 7x EX: log 2 3x + log 2 5y = log 2 15xy (since 3x*5y = 15xy)

4. The Quotient Property Definition: The log of a quotient can be expanded into the DIFFERENCE of the logs of the factors log b m/n = log b m – log b n (EXPANDING) EX: log 3 7/x = log 3 7 – log 3 x EX: log 2 3/5 = log 2 3 – log 2 5

5. The Quotient Property Definition: The DIFFERENCE of logs with the same base can be CONDENSED into the log of the fraction log b m – log b n = log b m/n (CONDENSING) EX: log 3 7 – log 3 2 = log 3 7/2 EX: log 2 3y – log 2 5x = log 2

6. The Power Property Definition: The log of a power expression can be expanded into the exponent times the log of the base log b m p = p ● log b m (EXPANDING) EX: log 3 x 5 = 5 log 3 x EX: log 3 11 = 11 log 3

7. The Power Property Definition: A number times the log of an expression can be CONDENSED into the log of the expression to the power of the number p ● log b m = log b m p (CONDENSING) EX: 5 log 3 x = log 3 x 5 EX: w log 3 = log 3 w

8. Additional Examples: Expand the logarithms (completely): 1.log 3x 2 = log 3 + log x 2 (Product Property) = log 3 + 2 log x (Power Property) 2.log 4x 5 y 7 = log 4 + log x 5 + log y 7 (Product) = log 4 + 5 log x + 7 log y (Power) 3. log = log (5y 4 ) – log (2x 3 ) (Quotient) = log 5 + log y 4 – log 2 – log x 3 (Product) = log 5 + 4 log y – log 2 – 3 log x (Power) (Why does the “3 log x” have to be subtracted?) TIP: Always do PRODUCT & QUOTIENT before POWER when expanding

9. Additional Examples: Condense the logarithms (completely): 1.log 6 + 4 log x = log 6 + log x 4 (Power Property) = log 6x 4 (Product Property) 2.log 17 + 2 log x + 0.5 log y = log 17 + log x 2 + log y 0.5 (Power) = log 17x 2 y 0.5 (Product) 3. log 7 + 2 log w – 3 log 2 – 4 log x = log 7 + log w 2 – log 2 3 – log x 4 (Power) = log (Product & Quotient Properties) TIP: Always do POWER before PRODUCT & QUOTIENT when condensing

10. PRACTICE: Expand the following logarithms. 1.log 4 7x 2.log 2 3.log x 7 4.log 5 3x 2 5. log 2 Write each problem on a sheet of paper and then either expand or condense… Condense the following logarithms. 6. log 6 – log 2 7. log 2 5 + log 2 x 8. 6 log 4 x 9. log 4 x + log 4 y – log 4 w 10. 4 log 3 x + 2 log 3 y