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Section 4.3 Logarithmic Functions

Chapter 4 – Exponential and Logarithmic Functions. Section 4.3 Logarithmic Functions. Exponential Functions. Recall from last class that every exponential function f ( x ) = a x with a >0 and a  1 is a one-to-one function and therefore has an inverse function.

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Section 4.3 Logarithmic Functions

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  1. Chapter 4 – Exponential and Logarithmic Functions Section 4.3 Logarithmic Functions 4.3 - Logarithmic Functions

  2. Exponential Functions Recall from last class that every exponential function f (x) = ax with a >0 and a 1 is a one-to-one function and therefore has an inverse function. That inverse function is called the logarithmic function with base a and is denoted by loga. 4.3 - Logarithmic Functions

  3. Definition • Logarithmic Function Let a be a positive number with a 1. The logarithmic function with base a, denoted by loga, is defined by logax = y  ay = x So logaxis the exponent to which the base a must be raised to give x. 4.3 - Logarithmic Functions

  4. Switching Between Logs & Exp. • NOTE: logax is an exponent! 4.3 - Logarithmic Functions

  5. Example – pg. 322 #5 • Complete the table by expressing the logarithmic equation in exponential form or by expressing the exponential equation into logarithmic form. 4.3 - Logarithmic Functions

  6. Properties of Logarithms 4.3 - Logarithmic Functions

  7. Example – pg. 322 • Use the definition of the logarithmic function to find x. 4.3 - Logarithmic Functions

  8. Graphs of Logarithmic Functions • Because the exponential and logarithmic functions are inverses with each other, we can learn about the logarithmic function from the exponential function. Remember, 4.3 - Logarithmic Functions

  9. Graphs of Log Functions 4.3 - Logarithmic Functions

  10. Example – pg. 323 • Graph the function, not by plotting points or using a graphing calculator, but by starting from the graph of a logax function. State the domain, range, and asymptote. 53. 58. 4.3 - Logarithmic Functions

  11. Definitions • Common Logarithm The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: • Natural Logarithm The logarithm with base e is called the natural logarithm and is denoted by: 4.3 - Logarithmic Functions

  12. Note • Both the common and natural logs can be evaluated on your calculator. 4.3 - Logarithmic Functions

  13. Properties of Natural Logs 4.3 - Logarithmic Functions

  14. Example – pg. 322 • Find the domain of the function. 4.3 - Logarithmic Functions

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