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Introduction to Logarithmic Functions

Introduction to Logarithmic Functions. Unit 6: Exponential and Logarithmic Functions. Introduction to Logarithmic Functions. GRAPHS OF EXPONENTIALS AND ITS INVERSE. In Grade 11, you were introduced to inverse functions.

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Introduction to Logarithmic Functions

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  1. Introduction to Logarithmic Functions Unit 6: Exponential and Logarithmic Functions

  2. Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • In Grade 11, you were introduced to inverse functions. • Inverse functions is the set of ordered pair obtained by interchanging the x and y values. f(x) f-1(x)

  3. Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • Inverse functions can be created graphically by a reflection on the y = x axis. f(x) y = x f-1(x)

  4. Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • A logarithmic function is the inverse of an exponential function • Exponential functions have the following characteristics: Domain: {x є R} Range: {y > 0}

  5. Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • Let us graph the exponential function y = 2x • Table of values:

  6. Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • Let us find the inverse the exponential function y = 2x • Table of values:

  7. Introduction to Logarithmic Functions GRAPHS OF EXPONENTIALS AND ITS INVERSE • When we add the function f(x) = 2x to this graph, it is evident that the inverse is a reflection on the y = x axis f(x) f-1(x) f(x) f-1(x)

  8. base exponent exponent base Introduction to Logarithmic Functions FINDING THE INVERSE OF AN EXPONENTIAL • Next, you will find the inverse of an exponential algebraically • Write the process in your notes y = ax x = ay Interchange x  y x = ay • We write these functions as: y = logax x = ay y = logax

  9. Introduction to Logarithmic Functions FINDING THE INVERSE OF AN EXPONENTIAL x y log a y x = Inverse of the Exponential Function Logarithmic Form Exponential Function

  10. The definition of a logarithm: A logarithm is an exponent! In fact, the logarithm of a number is the exponent to which a base must be raised to get the number. This is the inverse of raising a base to an exponent to get the number!

  11. Introduction to Logarithmic Functions CHANGING FORMS Example 1) Write the following into logarithmic form: a) 33 = 27 b) 44= 256 c) 27 = 128 d) (1/3)x=27 ANSWERS

  12. Introduction to Logarithmic Functions CHANGING FORMS Example 1) Write the following into logarithmic form: a) 33 = 27 log327=3 log4256=4 b) 45 = 256 c) 27 = 128 log2128=7 log1/327=x d) (1/3)x=27

  13. Introduction to Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log264=6 b) log255=1/2 c) log81=0 d) log1/39=-2 ANSWERS

  14. Introduction to Logarithmic Functions CHANGING FORMS Example 2) Write the following into exponential form: a) log264=6 26 = 64 b) log255=1/2 251/2 = 5 80 = 1 c) log81=0 (1/3)-2= 9 d) log1/39=2

  15. Introduction to Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log1/327 = x b) log5x = 3 c) logx(1/9) = 2 d) log3x = 0 ANSWERS

  16. Introduction to Logarithmic Functions EVALUATING LOGARITHMS Example 3) Find the value of x for each example: a) log1/327 = x b) log5x = 3 (1/3)x = 27 (1/3)x = (1/3)-3 x = -3 53 = x x = 125 c) logx(1/9) = 2 d) log3x = 0 x2 = (1/9) x = 1/3 30 = x x = 1

  17. Introduction to Logarithmic Functions BASE 10 LOGS Scientific calculators can perform logarithmic operations. Your calculator has a LOG button. This button represents logarithms in BASE 10 or log10 Example 4) Use your calculator to find the value of each of the following: a) log101000 b) log 50 c) log -1000 = Out of Domain = 3 = 1.699

  18. Introduction to Logarithmic Functions HOME FUN Pg451 : 1,5 – 11

  19. Let’s look at a few more graphs: Each graph represents an exponential function.

  20. The question is: Which equation goes with which graph? y = 2x, y = 5x, and y = 10x ?? Think before clicking to the next slide!!

  21. Did you guess correctly? y = 10x y = 5x y = 2x

  22. Conclusion: y = 10x y = 5x y = 2x When the base of an exponential function increases, the y-values increase more quickly!! Ex. y = 10x increases more quickly than y = 2x.

  23. Let’s look at some logarithmic functions.

  24. Try again! Which equation goes with which graph? y = log2x, y = log 5x, and y = log10x?? Think before clicking to the next slide!!

  25. Did you guess correctly? y=log2x y=log5x y=log10x

  26. Conclusion: y=log2x y=log5x y=log10x When the base of a logarithmic function increases, the y-values increase more slowly!! Ex. increases slower than. y=log2x y=log10x

  27. Home Fun • Pg 457 # 1 – 3, 4 (ii) (iii) (vi) , 5 a b e, 9

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