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Understanding Logarithmic Functions and Population Predictions

This lesson explores the basics of logarithmic functions and their applications in real-world scenarios, such as predicting population growth. It begins with the population of India in 1996, using a growth rate to derive formulas for predictions in 2000 and 2010. The session includes examples and exercises on converting between exponential and logarithmic forms, evaluating logarithms, and solving logarithmic equations. This comprehensive approach equips students with essential skills for handling logarithmic concepts in various contexts, including science and mathematics.

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Understanding Logarithmic Functions and Population Predictions

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  1. Warm-Up: February 1, 2013 • The population of India was 952,108,100 in 1996 and was growing at a rate of 1.3% per year. • Write the expression that would predict the population in 2000. • Write the expression that would predict the population in 2010

  2. Homework Questions?

  3. Logarithmic Functions Section 6.3 Part 1

  4. Essential Questions • How can we convert between logarithmic and exponential forms?

  5. One-to-One Property of Exponents

  6. Logarithm • The logarithm of a number to a certain base is the exponent to which the base must be raised to equal the number.

  7. Example 1 • Evaluate each of the following logarithms: • A) log525 • B) log28 • C) log381

  8. You-Try #1 • Evaluate each of the following logarithms

  9. Use of Logarithms (don’t copy) • pH scale in chemistry • Decibels – noise levels • Richter scale – earthquake strength • Apparent magnitude (brightness) of a star • Musical intervals – semitones

  10. Converting Forms Exponential form Logarithmic form 103 = 10003 = log101000 Exponent Base Move the base to the other side of the equation

  11. Converting Forms • You can take the log (with same base) of both sides • Or you can make both sides exponents with the same base. • Logs and exponents of the same base “cancel”

  12. Example 2 • Convert to Logarithmic form

  13. You-Try #2 • Convert to Logarithmic form

  14. Example 3 • Convert to Exponential Form

  15. You-Try #3 • Convert to Exponential Form

  16. Evaluating Logarithms • To evaluate a logbx, ask yourself, “b to what exponent gives me x?”

  17. Example 4: Solve • Solve v=log100.001

  18. You-Try #4: Solve • Solve v=log41

  19. Assignment • Page 374 #12-35, 56-67

  20. Warm-Up: February 4, 2013 • Evaluate the following logarithms

  21. Homework Questions?

  22. Logarithmic Functions Section 6.3 Part 2

  23. Essential Question • How can we solve logarithmic equations?

  24. Solving for y • Evaluate the logarithm • We did this last class

  25. Solving for b • Convert to exponential form • Use a radical or fractional exponent to isolate b • Remember b>0

  26. Example 5 – Solve for b

  27. You-Try #5 – Solve for b

  28. Solving for x • Convert to exponential form • Evaluate the exponential

  29. Example 6 – Solve for x

  30. You-Try #6 – Solve for x

  31. Assignment • Page 375 #68-85

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