1. Quiz 1) Convert log 2 4 = x into exponential form 2) Convert 3 y = 9 into logarithmic form 3) Graph y = log 4 x 2 x = 4 log 3 9 = y y = log 4 x

2. Properties of Logarithms With logs there are ways to expand and condense them using properties Product Property: log a (c*d) = log a c + log a d Examples: log 4 (2x) log 8 (x 2 y 4 ) = log 4 2 + log 4 x = log 8 x 2 + log 8 y 4 Division (Quotient) Property: log a (c/d) = log a c – log a d Examples: log 4 (2/x) log 8 (x 2 /y 4 ) = log 4 2 – log 4 x = log 8 x 2 – log 8 y 4 When two numbers are multiplied together within a log you can split them apart using separate logs connected with addition When two numbers are divided within a log you can split them apart using separate logs connected with subtraction

3. Properties of Logarithms (continued) Power Property: log a (c x ) = x*log a c Examples: log 4 (x 2 ) log 8 (2 x ) = 2log 4 x = xlog 8 2 Examples using more than one property log 3 (c 2 /d 4 ) log 4 (5x 7 ) log 8 ((4x 2 )/y 4 ) = log 4 5 +log 4 x 7 = (log 8 4 + log 8 x 2 ) – log 8 y 4 When a number is raised to a power within a log you multiply the exponent to the front and multiply it by the log (bring the exponent out front) = 2log 3 c – 4log 3 d = (log 8 4 + 2log 8 x) – 4log 8 y = log 3 c 2 – log 3 d 4 = log 4 5 +7log 4 x

4. log 9 (6 3 *2 10 ) = 3log 9 6 + 10log 9 2 = log 9 6 3 + log 9 2 10 Try These Log 1/2 (4 -3 *5 (2/3) ) = -3log 1/2 4 – (2/3)log 1/2 5 = log 1/2 4 -3 – log 1/2 5 (2/3) log 3 ((1/2) 3 /(-2) -4 ) = 3log 3 (1/2) – -4log 3 (-2) = log 3 (1/2) 3 – log 3 (-2) -4 = 3log 3 (1/2) + 4log 3 (-2)

5. Quiz 1) Find: log 5 125 2) What two numbers would log 4 24 be between? 5 ? = 1255 1 = 55 2 = 255 3 = 125 4 1 = 44 2 = 164 3 = 64 So log 5 125 = 3 So log 4 24 is between 2 and 3 3) Use a calculator to find log 4 24 log 4 24 = (log(24))/(log(4)) = 2.929

6. Condensing logarithms (undoing the properties) = log 5 (6/y) log 9 5 + 7log 9 x log 5 6 – log 5 y log 2 12 – (7log 2 z + 2log 2 y ) = log 9 5 + log 9 x 7 = log 9 (5x 7 ) = log 2 12 – (log 2 z 7 + log 2 y 2 ) = log 2 12 – (log 2 (z 7 y 2 )) = log 2 (12 /(z 7 y 2 ))

7. Solve for x Since the base is the same we can set the pieces that we are taking the log of equal to each other. log 5 25 = 2log 5 x log 4 x = log 4 2 25 = x 2 We use the properties to condense the log – then solve for x log 5 25 = log 5 x 2 5 = x x = 2

8. Try These log 3 6 = log 3 3 + log 3 x 6 = 3x log 3 6 = log 3 (3x) 2 = x 3 (1/3)log 4 x = log 4 4 x (1/3) = 4 log 4 x (1/3) = log 4 4 x = 64 (()) 33