Содержание
- 2. Material What are Logarithms? Laws of indices Logarithmic identities
- 3. Exponents 20 = 1 21 = 2 22 = 2 x 2 = 4 23 =
- 4. Problem We want to know how many bits the number 456 will require when stored in
- 5. A simpler way Round 456 up to the smallest power of 2 that is greater than
- 6. A simpler way Much better, but we really don’t like the rounding up to the smallest
- 7. Logarithms
- 8. Logarithms We only know 456, lets compute log base 2 of 456 log2456 = 8.861… Rounding
- 9. Logarithms If x = yz then z = logy x
- 10. Logarithms and Exponents If x = yz then z = logy x e.g. 1000 = 103,
- 11. Logarithms and Exponents: general form From lecture 1) base index form: number = baseindex then index
- 12. Graphs of exponents
- 13. Graphs of logarithms
- 14. Log plot
- 15. Three ‘special’ types of logarithms Common Logarithm: base 10 Common in science and engineering Natural Logarithm:
- 16. Laws of indices 1) a0 = 1 2) a1 = a
- 17. Laws of indices 1) a0 = 1 2) a1 = a Examples: 20 = 1 100
- 18. Laws of indices 1) a0 = 1 2) a1 = a Examples: 21 = 2 101
- 19. Laws of indices 3) a-x = 1/ax
- 20. Laws of indices 3) a-x = 1/ax Example: 3-2 = 1/32 = 1/27
- 21. Laws of indices 4) ax · ay = a(x + y) (a multiplied by itself x
- 22. Laws of indices 4) ax · ay = a(x + y) 42 · 43 = 4(2+3)
- 23. Laws of indices 5) ax / ay = a(x - y) 105 / 103 = 10(5-3)
- 24. Laws of indices X times X times X times y times
- 25. Laws of indices 6) (ax)y = axy (103)2 = 10(3x2) = 106 1,0002 = 1,000,000 (24)2
- 26. Laws of indices 7) ax/y = y√ax 10(4/2) = 2√104 102 = 2√10,000 = 100 2(9/3)
- 27. Logarithmic identities ‘Trivial’ Log form Index form logb 1 = 0 b0 = 1 logb b
- 28. Logarithmic identities 2 y · logb x = logb xy (bx)y = bxy
- 29. Logarithmic identities 2 examples y · logb x = logb xy (bx)y = bxy Examples: 9
- 30. Logarithmic identities 3 Negative Identity -logb x = logb (1/x) b-x = 1/bx Addition logb x
- 31. Negative Identity Taking log from both sides of the equation
- 32. Negative identity Negative Identity -logb x = logb (1/x) b-x = 1/bx Examples: -3 = -log2
- 33. Addition identity Taking log from both sides of the equation bx · by = b(x +
- 34. Addition identity examples Addition logb x + logb y = logb xy bx · by =
- 35. Subtraction Identity Taking log from both sides of the equation bx · by = b(x +
- 36. Subtraction identity examples Subtraction logb x - logb y = logb x/y bx / by =
- 37. Changing the base logb x = logy x / logy b leads to logb x =
- 38. Changing the base, examples 1 logb x = logy x / logy b Examples: 2 =
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