1. Exponentialand logarithmicfunctions Yr 11 maths methods

2. Objectives for Term 2 • To define and understand exponential functions. • To sketch graphs of the various types of exponential functions. • To understand the rules for manipulating exponential and logarithmic expressions. • To solve exponential equations. • To evaluate logarithmic expressions. • To solve equations using logarithmic methods. • To sketch graphs of functions of the form y = logax and simple transformations of this. • To understand and use a range of exponential models. • To sketch graphs of exponential functions. • To apply exponential functions to solving problems.

3. Introduction • Functions in which the independent variable is an index number are called indicial or exponential functions. For example: • f (x) = ax where a > 0 and a ≠ 1 • quantities which increase or decrease by a constant percentage in a particular time can be modelled by an exponential function. • Exponential functions can be seen in everyday life for example in science and medicine (decay of radioactive material, or growth of bacteria like those shown in the photo), and finance ( compound interest and reducing balance loans).

4. Index laws

5. Multiplication • When multiplying two numbers in index form with the same base, add the indices. • For example, 23 × 24 =(2 × 2 × 2) × (2 × 2 × 2 × 2) = 27 am × an = am+ n

6. Division • When dividing two numbers in index form with the same base, subtract the indices. am ÷ an = am- n

7. Raising to a power • To raise an indicial expression to a power, multiply the indices. (am)n = am × n = amn

8. Raising to the power of zero • Any number raised to the power of zero is equal to one. a0 = 1, a ≠ 0

9. Products and quotients

10. Remember

11. Questions

12. Answers (a)

13. Answers (b)

14. Answers (c)

15. Answers (d)

16. Homework • Page 220 Questions 1 – 3

17. More Questions

18. Answer without using your Cauculators

19. Answer with your calculators

20. Questions

21. Answer (a)

22. Answer (b)

23. Question

24. Answer

25. Homework • Page 220 – 221 - Questions 4 – 10

26. negative and rational powers

27. negative powers

28. Examples

29. Answer A

30. Answer B

31. Rational powers

32. Examples

33. Examples

34. Indicial equations

35. Indicial equations

36. Examples

37. Answer A

38. Answer B

39. Answer C

40. Solve the following

41. Answer

42. Answer

43. Graphs of exponential functions

44. Graphs of exponential functions

45. The effect of changing the “a” coeff

46. The effect of changing the “a” coeff

47. The effect of changing the “a” coeff

48. Reflections of exponential functions

49. Reflections of exponential functions

50. Reflections of exponential functions