Unit 11: Logarithms, day 3 3 properties to Expand and 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: log3 7x = log3 7 + log3 x log2 15 = log2 3 + log2 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: log3 7 + log3 x = log3 7x log2 3x + log2 5y = log2 15xy (since 3x*5y = 15xy)

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: log3 7/x = log3 7 – log3 x log2 3/5 = log2 3 – log2 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: log3 7 – log3 2 = log3 7/2 log2 3y – log2 5x = log2

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: log3 x5 = 5 log3 x 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 log3 x = log3 x5 w log 3 = log 3w

Additional Examples: Expand the logarithms (completely): TIP: Always do PRODUCT & QUOTIENT before POWER when expanding Expand the logarithms (completely): log 3x2 = log 3 + log x2 (Product Property) = log 3 + 2 log x (Power Property) log 4x5y7 = log 4 + log x5 + log y7 (Product) = log 4 + 5 log x + 7 log y (Power) 3. log = log (5y4) – log (2x3) (Quotient) = log 5 + log y4 – log 2 – log x3 (Product) = log 5 + 4 log y – log 2 – 3 log x (Power) (Why does the “3 log x” have to be subtracted?)

Additional Examples: Condense the logarithms (completely): TIP: Always do POWER before PRODUCT & QUOTIENT when condensing Condense the logarithms (completely): log 6 + 4 log x = log 6 + log x4 (Power Property) = log 6x4 (Product Property) log 17 + 2 log x + 0.5 log y = log 17 + log x2 + log y0.5 (Power) = log 17x2y0.5 (Product) 3. log 7 + 2 log w – 3 log 2 – 4 log x = log 7 + log w2 – log 23 – log x4 (Power) = log (Product & Quotient Properties)

PRACTICE: log4 7x log2 log x7 log5 3x2 5. log2 Write each problem on a sheet of paper and then either expand or condense… Condense the following logarithms. 6. log 6 – log 2 7. log2 5 + log2 x 8. 6 log4 x 9. log4 x + log4 y – log4 w 10. 4 log3 x + 2 log3 y Expand the following logarithms. log4 7x log2 log x7 log5 3x2 5. log2