1. Properties of Logarithms 3 properties to Expand and Condense Logarithmic Expressions; 1 formula to Change the Base Monday, February 8, 2016

2. Warm-Up Define the following terms: p.298 [TEXT] – Change of Base Formula – Product Property – Quotient Property – Power Property

3. Essential Question How are the properties of logarithms used to expand, condense & evaluate logarithmic expressions?

4. 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) Note: The BASE of the Logarithm must be and must stay the same when using this property.

5. 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)

6. The Quotient Property

8. The Power Property

10. 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

11. 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

12. The Change of Base Formula The Change of Base Formula can be used to change any single logarithm into the division of two logarithms of any desired base. log b x = log a (x)/ log a (b) … where “a is the desired base Ex: log 2 (7) = log (7)/log (2) … common log or = ln (7)/ ln(2) … natural log or= log 5 (7)/log 5 (2)… base 5 This makes evaluating logarithms of different bases easier. You can use the LOG or LN button on your calculator… log 2 (7) = log (7)/log (2) ≈ 2.8074

13. HOMEWORK Use your workbook pages 258-259 and do problems (1 – 6)

14. Reflection What is one bit of advice you would tell someone in another class who hasn't learned this yet?