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Activity 37

Activity 37. Logarithmic Functions (Section 5.2, pp. 398-405). Definition:. Let a be a positive number with a ≠ 1. The logarithmic function with base a, denoted by log a , is defined by. Properties of Logarithms (3 step proofs):. Let a be a positive number with a ≠ 1. Example 1:.

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Activity 37

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  1. Activity 37 Logarithmic Functions (Section 5.2, pp. 398-405)

  2. Definition: Let a be a positive number with a ≠ 1. The logarithmic function with base a, denoted by loga, is defined by

  3. Properties of Logarithms (3 step proofs): Let a be a positive number with a ≠ 1

  4. Example 1: Change each exponential expression into an equivalent expression in logarithmic form:

  5. Example 2: Change each logarithmic expression into an equivalent expression in exponential form:

  6. Example 3: Evaluate each of the following expressions: Property 3 Property 3 Property 3

  7. Property 4 Property 3 Property 1

  8. Graphs of Logarithmic Functions:

  9. Example 4: Find the domain of the function and sketch its graph.

  10. Common Logarithms: The logarithm with base 10 is called the common logarithm and is denoted by omitting the base: log x := log10 (x).

  11. Example 5 (Bacteria Colony): A certain strain of bacteria divides every three hours. If a colony is started with 50 bacteria, then the time t (in hours) required for the colony to grow to N bacteria is given by Find the time required for the colony to grow to a million bacteria.

  12. Definition: Natural Logarithms The logarithm with base e is called the natural logarithm and is denoted by ln: ln x := loge x. We recall again that, by the definition of inverse functions, we have

  13. Properties of Natural Logarithms:

  14. Example 6: Evaluate each of the following expressions: Property 3 Property 3 Property 4

  15. Example 7: Graph the function

  16. Example 8: Find the domain of the function

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