Unit 11 Logarithms day 3 3 properties to

The Product Property • Definition: The log of a product can be expanded into

The Product Property • Definition: The SUM of logs with the same base can

The Quotient Property • Definition: The log of a quotient can be expanded into

The Quotient Property • Definition: The DIFFERENCE of logs with the same base can

The Power Property • Definition: The log of a power expression can be expanded

The Power Property • Definition: A number times the log of an expression can

Additional Examples: TIP: Always do PRODUCT & QUOTIENT before POWER when expanding Expand the

Additional Examples: TIP: Always do POWER before PRODUCT & QUOTIENT when condensing Condense the

PRACTICE: Write each problem on a sheet of paper and then either expand or

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

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

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

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

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

The Power Property • Definition: The log of a power expression can be expanded into the exponent times the log of the base logb mp = p ● logb m (EXPANDING) EX: log 3 x 5 = 5 log 3 x EX: log 311 = 11 log 3

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 ● logb m = logb mp (CONDENSING) EX: 5 log 3 x = log 3 x 5 EX: w log 3 = log 3 w

Additional Examples: TIP: Always do PRODUCT & QUOTIENT before POWER when expanding Expand the logarithms (completely): 1. log 3 x 2 = log 3 + log x 2 (Product Property) = log 3 + 2 log x (Power Property) 2. log 4 x 5 y 7 = log 4 + log x 5 + log y 7 (Product) = log 4 + 5 log x + 7 log y (Power) 3. log = log (5 y 4) – log (2 x 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? )

Additional Examples: TIP: Always do POWER before PRODUCT & QUOTIENT when condensing Condense the logarithms (completely): 1. log 6 + 4 log x = log 6 + log x 4 (Power Property) = log 6 x 4 (Product Property) 2. log 17 + 2 log x + 0. 5 log y = log 17 + log x 2 + log y 0. 5 (Power) = log 17 x 2 y 0. 5 (Product) 3. log 7 + 2 log w – 3 log 2 – 4 log x = log 7 + log w 2 – log 23 – log x 4 = log (Power) (Product & Quotient Properties)

PRACTICE: Write each problem on a sheet of paper and then either expand or condense… Expand the following logarithms. Condense the following logarithms. 1. log 4 7 x 2. log 2 3. log x 7 4. log 5 3 x 2 5. log 2 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