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Find x, y, and z

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Find x, y, and z

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  1. Notes Day 8.2 PAP Algebra 2Objective: TLW…develop the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses (2A.11.A)Use parent functions to describe effects on graphs of exponential and logarithmic functions (2A.11.B)Examine asymptotic behavior (2A.11.B)Solve exponential/logarithmic equations using algebraic methods (2A.11.D)

  2. Find x, y, and z 2x = 4 2y = 8 2z = 6 Z = ? X = 2 y = 3 Mathematicians use a LOGARITHM to find z and we will study logarithmic functions this unit Exponential A logarithm is the inverse of an ______________ function

  3. f-1x = log2x f(x) = 2x 1/8 1/8 Note exp fcn has H.A. and log fcn has V.A. 1/4 1/4 1/2 1/2 1 1 2 2 4 4 8 8 Inverses These 2 graphs are reflections over the line _________ y = x

  4. ConvertExponentials and Logarithms If x = by then ________ log bx = y Look at the log function graph: x is ____________________ domain is _________ always greater than 0 Value of asymptote x > 0

  5. Convert the Exponential Equations to Logarithms If x = b y then ________ 1. 2.3. 32 = 9 log b x = y log 3 9 = 2 log 10 100 = 2 log 2 16 = 4 Note that we are changing form …. not solving

  6. Convert the Exponential Equations to Logarithms If x = b y then ________ 4. 5. 6. 1 = 50 log b x = y log 10 0.1 = -1 log 5 1 = 0 log 4 = -2

  7. Write the Logarithmic Equations in Exponential Form If x = b y then ________ 7. 8. 9. log b x = y log 8 64 = 2 log 100 = 2 log 2 8 = 3 When no base is written ….it is a common log with base 10

  8. Evaluate each Logarithm Now we are solving for x If x = b y then ________ 1. 2. =x 3. log1000 = x y = log b x log3 27 = x log6

  9. Evaluate each Logarithm If x = b y then ________ 4. 5. =x 6. log816 =x y = log b x log½ log 9 27 = x

  10. Special Logarithm Values 1 x 0 logb1=_____ logbb=_____ logbbx=_____ b x= 1 b x= b b x= b x Why are these good rules to know: (not on your notes) Find the y-intercept of Substitute 0 for x (0,4)

  11. A common logarithm is a logarithm that uses base ______ 10 For example: log x = _____________ (The log key on the calc. is the common log) log10x

  12. change of base formula Remember on the first slide when we wanted to solve Example: log 2 7 = Use the change of base Formula: log b x = So now solve

  13. Translating Logarithmic Functions Parent Function: The k Vertical Shift: The h Horizontal Shift: Stretch/Compress: Reflection in x-axis:

  14. On an earlier slide we graphed an exponential function and its inverse. This current slide is not in your notes – but lets prove why y=2x and y = log 2x are inverses. y = log 2 x Switch variables to find inverse equations x = log 2 y Convert from log to exp. form

  15. Graph Look above at the parentfunction of y = log2x Horiz shift ________ Vert Shift = _______ V Asymptote: ______ Domain: __________ Left 1 4 Up 4 x = -1 x > -1 1 X-intercept:______ 0 = log 2 (x+1) +4 – 4 = log 2 (x+1) 2 -4 = x+1

  16. Activity: Now lets see what you know. I will show you some problems. When I ask for the answer, please show the color of the matching correct answer. HW : WS 8.2 – which is is due next class. We will also be taking a quiz next class on these concepts.

  17. Express 24=16 in Logarithm Form. • A. log24=16 • B. log216=4 • C. log416=2 • D. log164=2

  18. Express ab=c in Logarithm Form. • A. logbc=a • B. logcb=a • C. logab=c • D. logac=b

  19. Express logab=c in Exponential Form. • A. bc=a • B. ac=b • C. ab=c • D. ba=c

  20. Express log39=2 in Exponential Form. • A. 39=2 • B. 23=9 • C. 32=9 • D. 92=3

  21. Evaluate log28 • A. 3 • B. 4 • C. 16 • D. 256

  22. Evaluate log (8) • A. -4 • B. -3 • C. 3 • D. 4

  23. Evaluate log (81) • A. -4 • B. -27 • C. 27 • D. 243

  24. How is the graph of f(x) = log3(x-1)-5transformed from its parent function A. Translated down 1 and left 5 B. Translated up 1 and left 5 C. Translated left 1 and down 5 D. Translated right 1 and down 5

  25. What is the domain of the graph f(x) = log3(x-1)-5 A. X 1 B. X 1 C. X -1 D. X -1

  26. What is the x-intercept of the graph f(x) = log3(x-1)-2 A. (7,0) B. (8,0) C. (9,0) D. (10,0)

  27. Solve for x: A. X=1/3 B. X=27 C. X=-2 D. X=-27

  28. Solve for x: A. X=-5 B. X=-3 C. X=3 D. X=7

  29. Solve for x: A. X=1.5 B. X=5 C. X=6 D. X=9

  30. Solve for x: A. X=10/3 B. X=4 C. X=16 D. X=64

  31. Solve for x: A. X=-81 B. X=9 C. X=2/3 D. X=3/2

  32. Solve for x: A. X=-27 B. X=-9 C. X=-4 D. X=27

  33. Quiz Next Class: • Like HW 8.2

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