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3-2 Logarithmic Functions

3-2 Logarithmic Functions. Chapter 3 Exponential and Logarithmic Functions. Warm-up. Describe the transformation of f(x) that results in the graph of g(x). Then sketch each graph. 1. 2. 3. .

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3-2 Logarithmic Functions

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  1. 3-2 Logarithmic Functions Chapter 3 Exponential and Logarithmic Functions

  2. Warm-up • Describe the transformation of f(x) that results in the graph of g(x). Then sketch each graph. 1. 2. 3.

  3. Key Concept: Relating Logarithmic and Exponential Forms • Exponential Form • Logarithmic Form base exponent base exponent

  4. When evaluating a logarithm, remember that the logarithm is the exponent. • To evaluate a logarithm, change it to exponential form first. Then use what is known about exponents to simplify.

  5. Example 1: Evaluate each logarithm a. b. c. d.

  6. Key Concept: Basic Properties of Logarithms

  7. Example 2: Apply the Properties of Logarithms Evaluate each expression: 1. 2. 3. 4.

  8. Key Concept: Basic Properties of Common Logarithms • A logarithm with base 10 or log10 is called a common logarithm, and it is often written without the base. • The common logarithm function y = log x is the inverse of the exponential function y = 10x. • The properties of common logarithms also hold true for common logarithms.

  9. Key Concept: Basic Properties of Common Logarithms

  10. Example 3: Evaluate each expression • log 10,000 • 10log 12 • log 14 (use a calculator) • log (-11)

  11. Key Concept: Basic Properties of Natural Logarithms • A logarithm with the base e or loge is called a natural logarithmand is denoted ln. • The natural logarithm function y = ln x is the inverse of the exponential function y = ex. • The properties for logarithms also hold true for natural logarithms.

  12. Key Concept: Basic Properties of Natural Logarithms • ln 1 = 0 • ln e = 1 • ln ex = x • e ln x = x

  13. Example 4: Evaluate each expression • ln e 4.6 • ln (-1.2) • e ln 4 • ln 7

  14. Graphs of Logarithmic Functions?

  15. Real World Example: Earthquakes • Richter Scale

  16. Real World Example: Earthquakes • Richter Scale amplitude

  17. Real World Example: Earthquakes • Richter Scale amplitude Period of the seismic wave in seconds

  18. Real World Example: Earthquakes A factor that accounts for the weakening of seismic waves • Richter Scale amplitude Period of the seismic wave in seconds

  19. Real World Example: Earthquakes A factor that accounts for the weakening of seismic waves • Richter Scale 1. Find the intensity of an earthquake with an amplitude of 250 microns, a period of 2.1 seconds, and B = 5.4. amplitude Period of the seismic wave in seconds

  20. Real World Example: Earthquakes A factor that accounts for the weakening of seismic waves • Richter Scale 2. Earthquakes with an intensity of 6.1 or greater can cause considerable damage. Determine the amplitude of an earthquake whose intensity is 6.1 with a period of 3.5 seconds and B = 3.7. amplitude Period of the seismic wave in seconds

  21. Assignment: p. 178-9 • 1 – 23 odds, 27 Be sure to show the set-up used to calculate this one.

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