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8.7 – Natural Logarithms

8.7 – Natural Logarithms. Natural Logarithm. Natural Logarithm – log e. Natural Logarithm – log e = ln. Natural Logarithm – log e = ln e. Natural Logarithm – log e = ln e ≈ 2.7183. Natural Logarithm – log e = ln e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e 2.

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8.7 – Natural Logarithms

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  1. 8.7 – Natural Logarithms

  2. Natural Logarithm

  3. Natural Logarithm – loge

  4. Natural Logarithm – loge = ln

  5. Natural Logarithm – loge = ln • e

  6. Natural Logarithm – loge = ln • e ≈ 2.7183

  7. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e2

  8. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd

  9. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd , ex

  10. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd , ex, 2

  11. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd , ex, 2, Enter

  12. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd , ex, 2, Enter e2 ≈ 7.3891

  13. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd , ex, 2, Enter e2 ≈ 7.3891 b. ln 4

  14. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd , ex, 2, Enter e2 ≈ 7.3891 b. ln 4 ln

  15. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd , ex, 2, Enter e2 ≈ 7.3891 b. ln 4 ln, 4

  16. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd , ex, 2, Enter e2 ≈ 7.3891 b. ln 4 ln, 4, Enter

  17. Natural Logarithm – loge = ln • e ≈ 2.7183 Ex. 1 Evaluate each expression. a. e22nd , ex, 2, Enter e2 ≈ 7.3891 b. ln 4 ln, 4, Enter ln 4 ≈ 1.3863

  18. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5

  19. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5 by = x

  20. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5 by = x logbx = y

  21. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5 by = x logbx = y loge5 = p

  22. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5 by = x logbx = y loge5 = p ln 5 = p

  23. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5 b. ln c = 7 by = x logbx = y loge5 = p ln 5 = p

  24. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5 b. ln c = 7 by = x logec = 7 logbx = y loge5 = p ln 5 = p

  25. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5 b. ln c = 7 by = x logec = 7 logbx = y logbx = y loge5 = p ln 5 = p

  26. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5 b. ln c = 7 by = x logec = 7 logbx = y logbx = y loge5 = p by = x ln 5 = p

  27. Ex. 2 Write an equivalent exponential or logarithmic equation. a. ep= 5 b. ln c = 7 by = x logec = 7 logbx = y logbx = y loge5 = p by = x ln 5 = p e7= c

  28. Ex. 3 Evaluate each expression. a. eln 7

  29. Ex. 3 Evaluate each expression. a. eln 7

  30. Ex. 3 Evaluate each expression. a. eln 7 = 7

  31. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. ln e4x + 3

  32. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. lne4x + 3

  33. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. ln e4x + 3 = 4x + 3

  34. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. ln e4x + 3 = 4x + 3 Ex. 4 Evaluate each expression. a. 5e-x– 7 = 2

  35. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. ln e4x + 3 = 4x + 3 Ex. 4 Evaluate each expression. a. 5e-x– 7 = 2 5e-x= 9

  36. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. ln e4x + 3 = 4x + 3 Ex. 4 Evaluate each expression. a. 5e-x– 7 = 2 5e-x= 9 e-x= 1.8

  37. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. ln e4x + 3 = 4x + 3 Ex. 4 Evaluate each expression. a. 5e-x– 7 = 2 5e-x= 9 e-x= 1.8 lne-x= ln 1.8

  38. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. ln e4x + 3 = 4x + 3 Ex. 4 Evaluate each expression. a. 5e-x– 7 = 2 5e-x= 9 e-x= 1.8 ln e-x= ln 1.8 -x = ln 1.8

  39. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. ln e4x + 3 = 4x + 3 Ex. 4 Evaluate each expression. a. 5e-x– 7 = 2 5e-x= 9 e-x= 1.8 ln e-x= ln 1.8 -x = ln 1.8 x = -ln 1.8

  40. Ex. 3 Evaluate each expression. a. eln 7 = 7 b. ln e4x + 3 = 4x + 3 Ex. 4 Evaluate each expression. a. 5e-x– 7 = 2 5e-x= 9 e-x= 1.8 ln e-x= ln 1.8 -x = ln 1.8 x = -ln 1.8 ≈ -0.5878

  41. b. ln 5x = 4

  42. b. ln 5x = 4 eln5x = e4

  43. b. ln 5x = 4 eln5x = e4 5x = e4

  44. b. ln 5x = 4 eln5x = e4 5x = e4 x = e4 5

  45. b. ln 5x = 4 eln5x = e4 5x = e4 x = e4 5 x ≈ 10.9196

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