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8.4

8.4. Logarithmic Functions. Learning Targets. Students should be able to… Evaluate logarithmic functions. Graph logarithmic functions. Warm-up. Simplify the expression. Homework Check.

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8.4

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  1. 8.4 Logarithmic Functions

  2. Learning Targets • Students should be able to… • Evaluate logarithmic functions. • Graph logarithmic functions.

  3. Warm-up • Simplify the expression

  4. Homework Check

  5. You know that 22 = 4 and 23 = 8. But do you know the value of x that satisfies the equation 2x = 6? • We can guess that the value of x is between 2 and 3, but to find the exact value, we use something called a logarithm.

  6. Definition of Logarithm with Base b If bx = y, then logby = x. x = exponent b = base Read as: log base b of y

  7. Example 1: Write the equation in EXPONENTIAL FORM. a) log3 9 = 2 b) log8 1 = 0 c) log5 1/25 = -2 d) log1/2 2 = -1 32 = 9 80 = 1 ½-1 = 2 5-2 = 1/25

  8. Special Logarithm Values- Let b be a positive real number such that b is not 1. • Logarithm of base b • Logb b = 1 because b 1 = b • Logarithm of 1 • Logb 1 = 0 because b 0 = 1

  9. Example 2: Evaluate each expression. a) log4 16 b) log2 8 c) log 4256 d) log2 2 2 3 4 1

  10. Example 2: Evaluate each expression. a) log64 8 b) log32 2 c) log 1/4256 d) log2 1/8 1/5 1/2 -3 -4

  11. Common Logarithm: log with base 10 • Notation: log x • If no base is written, assume the base is 10. • Natural Logarithm: log with base e • Notation: ln x

  12. Example 3: Use your calculator to evaluate. a) log 7 b) ln 0.25 c) ln 0.1 d) log 10 ≈.845 ≈ -1.386 ≈ -2.303 = 1

  13. Finding Inverses Example 5: Find the inverse of each function. a) y = log4 x b) y = ln (x + 1) c) y = log (x - 3) Find the inverse of y = logb x x = ln(y+1) ex = y + 1 y = ex – 1 x = log(y – 3) 10x = y – 3 y = 10x + 3 x = log4y y = 4x y = bx

  14. Notice: Exponential and Logarithmic functions are inverses! Using Inverses to Simplify Expressions Let f(x) = bx and let g(x) = logb x. Since the two funtions are inverses, f(g(x)) and g(f(x)) should both equal what? Find f(g(x)) and g(f(x)). f(g(x) = blogbx = x g(f(x)) = logbbx = x

  15. Example 6: Simplify each expression. a) b) c) d) x x x 2x

  16. Remember when graphing exponential functions: • Plug in 0 and 1 for x. • Domain is all real numbers • There is a horizontal asymptote at y = k • Since logarithmic functions are inverses of exponential functions, what do you expect for the logarithmic graphs?

  17. Graphing logarithmic functions The graph of y = logb (x – h) + k has the following characteristics: • x = h is a vertical asymptote • The domain is x > h, the range is all real numbers • If b > 1, the graph moves up and right • If 0 < b < 1, the graph moves down and right

  18. Example 7: Graph each function. State the domain and range. • y = log2x 1. Sketch in asymptote x = 0 2. This is an unshifted graph pick 1 and the base to plug in for x

  19. Example 7: Graph each function. State the domain and range. a) y = log5 (x + 1) 1. Sketch in asymptote x = -1 2. Look at the unshifted graph and pick two points that will result in integers y = log5x Domain: x > -1 Range: all real Shift left 1

  20. b) y = log3 x + 2 1. Sketch in asymptote x = 0 2. Look at the unshifted graph and pick two points that will result in integers y = log3x Domain: x > 0 Range: all real Shift up 2

  21. b) y = log1/4 (x – 2) – 3 1. Sketch in asymptote x = 2 2. Look at the unshifted graph and pick two points that will result in integers y = log1/4x Domain: x > 2 Range: all real Shift right 2 and down 3

  22. Extra Problem not in Guided Notes b) y = log1/2 (x + 4) + 2 1. Sketch in asymptote x = -4 2. Look at the unshifted graph and pick two points that will result in integers y = log1/2x Domain: x > - 4 Range: all real Shift left 4 and up 2

  23. Work on Extra Problems in Guided Notes Book! • See how the changes in the equation impact the graph.

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