1. Logarithms and Exponentials
Mathematics Content

2. On the Reference sheet for this topic:
Main Ideas:
review of index laws
what is a logarithm?
relationship between y = ax and y = logax; change of base
derivatives of logs and exponentials
differentiation and integration of composite functions
On the Reference sheet for this topic:

3. Index Laws Review

4. Cambridge

8. What is e?
Like π, e is an approximation of an infinite decimal. It is between 2 and 3, (around ), and ex has the property that ex has a gradient of 1 at (0,1). It is called ‘Euler’s number’ after Leonard Euler, and has been found to decimal places!

9. Transformations of Exponential graphs
The red curve is y=ex. The blue curve is y=ex+3. Adding ‘3’ raises the y value by 3 (this is a ‘transformation’). Similarly, y=e(x-2) will move the curve 2 units to the right of y=ex.

10. Cambridge

13. Differentiating y = ex
Similar results occur for other exponential functions. In general,
d (ax) = kax where k is a constant dx

15. Groves

17. Function of a Function (Chain Rule)

19. Groves

21. Cambridge

27. Cambridge

32. Integrating Exponentials
Consider integrating as the opposite process as differentiating. It follows that:
and:

34. Groves

36. Cambridge

41. Logarithmic Functions
Logarithms and exponentials are directly related. Using logarithms allows us to change the subject of exponential equations and solve them more easily.
DEFINITION:
If y=ax, then x is called the logarithm of y to the base a

42. Relationship between exponentials and logarithms
The two graphs are reflections of each other, reflected across the line y=x, and their domains and ranges are exchanged.

43. We use two different type of logarithm - when you see ln notation it means logarithm using ‘logex’ while log is used for ‘log10x’.

45. Groves

48. Logarithm Laws

49. Other useful bits and pieces:

50. Given that log 5 3=0.68 and log 5 4=0.86, find:

51. Groves

53. Groves

54. Cambridge

59. Cambridge

61. Differentiating Logarithmic Functions

62. Function of a Function (Chain Rule)

65. Groves

67. Cambridge

71. Cambridge

74. Integration of Logarithmic Functions
As integration is the inverse of differentiation,

77. Groves

79. Cambridge

83. You should be able to:
integrate, differentiate and manipulate exponential and logarithmic functions
complete a summary of the main points in this topic
go back through other topics where we skipped questions involving this work
complete HSC questions on this topic