1. February 13, 2012 At the end of today, you will be able to graph a logarithmic function. Warm-up: 1. 2. 3. Describe the transformation for: f(x) = -3 x – 5 4. Solve: 4 3 = 8 5x – 2 HW 3.2: Pg. 236 #2- 16 even, 17-22 all, 31, 40-60even

2. Correct HW 3.1b Pg. 227 #45-52 all, 61-63 45. x = 252. x = 2, 3 46. x = 761. $222,822.57 47. x = -362. $212,605.41 48. x = -463. $35.45 49. x = 1/3 50. x = 5/2 51. x = 3, -1

3. 3.2 Logarithmic Graphs What does the inverse of an exponential graph look like? (Hint: Graphs of inverses are reflections over what line?) XY 01 12 24 38 y = 2 x y = x y = log 2 x XY 10 21 42 83 X and Y are switched! Y approaches 0! (x-axis asymptote) X approaches 0! (y-axis asymptote) Y - int X - int

4. Exponential Logarithmic To change from exponential to logarithmic form, use the formula below: log b x = y if and only if x = b y Example 1: log 2 8 = 3 b = 2, x = 8, y = 3 x = b y 8 = 2 3 Step 1: Identify x, y, and b from equation Step 2: Use x, y, and b to plug into other equation Step 3: Double check – does this make sense? Show them the log wheel!

5. log b x = y if and only if x = b y Step 1: Identify x, y, and b from equation OR use the Log Roll! Step 2: Use x, y, and b to plug into other equation Step 3: Double check – does this make sense? You Practice! 1) log 5 125 = 3 2)log 8 1 = 0 Rewrite in Log Form 3) 6 2 = 36 4) 2 5 = 32 *Hint: The log is always equal to the exponent.

6. Evaluating Logs Example 2: f(x) = log 2 x, if x = 16 1. Substitute x = 16 f(16) = log 2 16 2. Equate to y: log 2 16 = y 3.Rewrite in exponential form 2 y = 16 4. Solvey = 4 Practice: 1.f(x) = log 4 x, x = 64 2.f(x) = log 10 x, x = 1000 3.f(x) = log 6 x, x = 1 4.

7. Can’t we just put this in our calculator??? NO, the common log is log base 10, log 10. Log 10 is the only function on your calculator. For other logs, you will have to rewrite in exponential form.

8. How do we graph logarithms? Example 3: Graph f(x) = log 4 x log 4 x = y 4 y = x Step 1: Equate to y and change to exponential form Step 2: Make a table! Hint – Choose numbers for y, then find x! XY 0 1 2 1 4 16 Y – axis is the asymptote! X – int (1, 0) What is the domain and range?

9. Transformations with Logarithms Describe the Transformations for each Log function. 1.y = log (x + 5) 2.y = log x – 3 3.y = – log x 4.y = 2log x 5.y = log (x – 2) + 1 Left 5 Down 3 Reflect over x Stretch vertically Right 2, up 1

10. Summary for Log Functions: Graphing logs: y = log 3 x Describing the transformations based on the equations below: a)f(x) = log (x + 5) b) f(x) = 2logx + 1 Changing forms! (Exponent Logarithm) a)8 3 = 512 b) Evaluating Logs a) log 3 81

11. Base e and Natural log Log base e is called the natural log function. log e x = ln x The rules for e and natural log are the same. Example 4: Write the log equation in exponential form: ln 4 = 1.386… e 1.386… = 4