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8.4 Logarithmic Functions

8.4 Logarithmic Functions. Relationship to Exponential Function. Recall the exponential function. The inverse is. Log b x = y. Definition of a Logarithm with base b. Let b and y be positive numbers and b  1. The logarithm of y with base b is denoted by Log b y and is defined as

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8.4 Logarithmic Functions

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  1. 8.4 Logarithmic Functions

  2. Relationship to Exponential Function Recall the exponential function The inverse is Logb x = y

  3. Definition of a Logarithm with base b Let b and y be positive numbers and b  1. The logarithm of y with base b is denoted by Logb y and is defined as Logb y = x if and only if bx = y. Logarithmic Form Exponential Form Key: • Logb y and bx = y are equivalent • The base must be positive • The number that you are taking the log of must be positive  The value of the log is equal to the exponent.

  4. Change Logarithms to Exponential Form Log3 9 = 2 Log5 5 = 1 Log½4 = -2 Log19 1 = 0

  5. Change Exponential Equations to Logarithmic Form And Evaluate Example: Evaluate Log2 64 Evaluate Log25 5 Log6 1 Change to Exponential form

  6. Common Logarithm Log10 x = y is the common logarithm. Denoted simply as Log x Note: If you do not see a base written with the log, then the base is 10.

  7. Natural Logarithm Loge x = y is the natural log. Denoted as Ln y = x.

  8. Special Values of Logarithms Logb 1 = 0 because b0 = 1 Logb b = 1 because b1 = b Inverses Logb bx = x because g(f(x)) = x because f(g(x)) = x

  9. The Graph of Logarithmic Functionsy = logb (x - h) + k • x = h is the asymptote. • Domain x > h. • Range y is all real numbers. • If b > 1, the graph increases up to the right. • If 0 < b < 1, the graph reflects down. The graph decreases left to right.

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