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Exploring Logarithmic Functions: Definitions, Properties, and Examples

Learn the basics of logarithmic functions with detailed definitions, properties, and practical examples to enhance your understanding. Explore how to switch between logarithmic and exponential functions, graph logarithmic functions, and grasp common and natural logarithms.

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Exploring Logarithmic Functions: Definitions, Properties, and Examples

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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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