Logarithms of Products

2. The first property we discuss is related to the product rule for exponents: Logarithms of Products Lets examine log3(9 · 27) vs log39+ log327. Note that log3(9 · 27) = log3243 = 5 35 = 243 and that log39 + log327 = 2 + 3 = 5. 32 = 9 and 33 = 27 So log3(9 · 27) = log39+ log327.

3. Example Express as an equivalent expression that is a single logarithm: loga6 + loga7. Solution Using the product rule for logarithms loga6+ loga7 = loga(6 · 7) = loga(42).

4. The second basic property is related to the power rule for exponents: Logarithms of Powers

5. Example Use the power rule to write an equivalent expression that is a product: a) loga6–3; Solution Using the power rule for logarithms a) loga6-3= –3loga6 = log4x1/2 Using the power rule for logarithms = ½log4x

6. The third property that we study is similar to the quotient rule for exponents: Logarithms of Quotients

8. Example Example Express as an equivalent expression that is a single logarithm: Express as an equivalent expression that is a difference of logarithms: loga6 – loga7. log3(9/y). Solution Using the quotient rule for logarithms log3(9/y) = log39– log3y. Solution Using the quotient rule for logarithms “in reverse” loga6 – loga7 = loga(6/7)

9. Example Expand to an equivalent expression using individual logarithms of x, y, and z. Using the Properties Together Solution = log4x3 – log4 yz = 3log4x – log4 yz = 3log4x – (log4 y+log4z) = 3log4x – log4 y – log4z

10. Solution continued

11. Condense to an equivalent expression Example using a single logarithm. Solution = logbx1/3 – logby2 + logbz

12. Example Simplify: a) log668 b) log33–3.4 Solution 8 is the exponent to which you raise 6 in order to get 68. a) log668= 8 b) log33–3.4 = –3.4