Exponential equations and logarithms

1. Exponential equations and logarithms

2. Fun with exponential equations • Every pair share a packet of M&Ms. • You are supposed to firstly empty the M&Ms on the serviettes provided randomly. • Count the number of M&Ms in all and record your reading as the zeroth reading. Then eat all the M&Ms facing up (i.e. with the letters on top) and count those remaining as the first reading. • Now put the M&Ms back into the bag, shake and pour out again. • Eat those facing up and count the remaining number of M&Ms again and record your reading as the second reading. • Continue the procedure until you have no more M&Ms remaining.

3. Fun with exponential equations • Now plot these points on Microsoft Excel, following my instructions. • What kind of graph does your readings give? • The functions that gives this kind of graphs are called exponential functions.

4. Exponential equations • Exponential Growth • Exponential Decay

5. The Beggar and the King

6. The Beggar and the King • Do you think the beggar made a wise request? • Was the King right in agreeing to the beggar’s request? • What do you think will happen?

7. The Beggar and the King

8. Exponential Growth

9. The Beggar and the King • Moral of the story: • Know your Mathematics! Use them in everyday life! • 学以致用！

10. Folding a piece of paper • http://raju.varghese.org/articles/powers2.html

11. Exponential Decay of radioactive substances • Cesium-137 and strontium-90 present long-term environmental hazards and can be absorbed throughout the body, particularly bones. Plutonium-239 exposure often leads to lung cancer, and it has a half-life of 24,000 years, so it would be around for a long, long time. • (A half-life is the amount of time it takes for half of the radioactive isotopes in a substance to decay.)

12. Exponential decay

13. Graphs of exponential functions y = 3x y = 2x

14. Graphs of exponential functions

15. Graphs of exponential functions Looks the same? Why?

16. Logarithm Index Logarithmic form Index form Base For any positive number a, except 1, What you want to know is: a to the power of WHAT gives y? This WHAT is your x, which is what you want to find in the logarithmic form.

17. Logarithm • Why can’t a =1 or a < 0? • a is the base of the logarithm. It cannot be equal to 1 as 1xis always 1. a cannot be negative as powers of negative numbers change sign. • For logay to be defined: • y > 0, why? • a > 0, a ≠ 1 For any positive number a, except 1,

18. Logarithm • For logay to be defined: • y > 0 • a > 0, a ≠ 1 • The following are not defined: • log12 • log-34 • log2(-1)

19. Logarithm Convert to logarithmic form. What is the base here? What is the index here? Convert to index form. What is the base here? What is the index here?

20. Convert the following to logarithmic form Convert the following to index form

21. Solve the following equations

22. 2 important conclusion What conditions of a do you need to impose here?

23. Evaluate the following

24. The Common Logarithm • Logarithms with a base of 10 are called common logarithms • Use the LOG key on your calculator to find the values of common logarithms. • log10x is often abbreviated as lgx. • Uses: In chemistry as a measure of acidity, in earthquakes as a measure of the strength (Richter Scale) etc. For earthquakes, the wave amplitude is typically very big, so a common logarithm scale is used. Think: a 9.0 earthquake (Japan) is how many times more powerful (in terms of wave amplitude) than an 8.0 earthquake (Sichuan earthquake)?

25. Solve the following equations, using your calculators, giving your answers correct to 4 s.f. Skip first x = 6.459 x = 1.364 x = 1.883 or x=0.7384

26. Natural Logarithm • There is another logarithm to the base of a special irrational number called e, named after Leonhard Euler. • e has a value of 2.7183………. • Logarithm to the base e, logex, is often abbreviated as lnx. • lnx is called Natural logarithm, or Naperian logarithm (after John Napier) • Can you find e and lnon your calculator? • Where is it used? Radioactive decay, first order reaction in Chemistry, calculus etc.

27. Solve the following equations Skip first

28. Laws of logarithm Product Law Quotient Law Power Law Note: m and n are positive and a > 0, a ≠ 1

29. Proofs • Product Law:

30. Proofs • Quotient Law:

31. Proofs • Power Law:

32. Are the following true?

33. Simplify the following

34. Logarithmic equations of the same base • For two logarithms of the same base,

35. Solve the following equation

36. Solve the following equation

37. Solve the following simultaneous equations

38. Change of bases • If a, b and c are positive numbers and a≠1,c≠1, then:

39. Proof

40. Example • Evaluate log75 × log59 × log97 • Hint: Change all the bases to common log!

41. Example • Evaluate Hint: Change all the bases to common log! Ans: 8

42. Solve the following equation

43. Examples (using all you have learnt) Given that log43 = a and log45 =b, express log445 in terms of a and b.

44. Exercise • Exercise 2.4 • Qns 4e-h, 5d-f, 6, 7, 8, 9, 10 • Exercise 2.5 • Qns 2e-f, 3e-f, 4e-f • Exercise 2.6 • Qns 4c, 5d-f, 6a-d, 7, 8, 9, 10d-f, 11 • Exercise 2.7 • Qns 1e-h, 2, 3, 4a, 5c-e, 6c-d, 7, 8

45. Equations of the form ax=b • Remember these? • Can you solve the equations below like what you did?

46. Equations of the form ax=b • If you cannot express both sides of the equation with the same base, the strategy is to take LOGARITHM on both sides.

47. Solve the following equations:

48. Example x = 2.22

49. Graphs of y=e-x and y=ex

50. Graphs of lgx and lnx