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Transcript

  • 1. 4.3 Laws of ExponentsRomans 8:24-25 For in this hope we were saved.Now hope that is seen is not hope. For who hopes forwhat he sees? But if we hope for what we do notsee, we wait for it with patience.
  • 2. Remember that logs are exponents.Therefore, these laws should make sense ...
  • 3. Remember that logs are exponents.Therefore, these laws should make sense ...1) log a (AB) = log a A + log a B
  • 4. Remember that logs are exponents.Therefore, these laws should make sense ...1) log a (AB) = log a A + log a B ⎛ A ⎞2) log a ⎜ ⎟ = log a A − log a B ⎝ B ⎠
  • 5. Remember that logs are exponents.Therefore, these laws should make sense ...1) log a (AB) = log a A + log a B ⎛ A ⎞2) log a ⎜ ⎟ = log a A − log a B ⎝ B ⎠3) c log a (A) = c log a A
  • 6. Logs can be manipulated using these lawsto either expanded form or combined form.
  • 7. Expand1) log 3 (9x)
  • 8. Expand1) log 3 (9x) log 3 9 + log 3 x
  • 9. Expand1) log 3 (9x) log 3 9 + log 3 x 2 + log 3 x
  • 10. Expand2) log 7 (x y ) 3 4
  • 11. Expand2) log 7 (x y ) 3 4 3 4 log 7 x + log 7 y
  • 12. Expand2) log 7 (x y ) 3 4 3 4 log 7 x + log 7 y 3log 7 x + 4 log 7 y
  • 13. Expand ⎛ a b ⎞3) ln ⎜ 8 ⎟ ⎝ c ⎠
  • 14. Expand ⎛ a b ⎞3) ln ⎜ 8 ⎟ ⎝ c ⎠ 1 ⎛ ⎞ ln ⎜ ab ⎟ − ln ( c ) 2 8 ⎝ ⎠
  • 15. Expand ⎛ a b ⎞3) ln ⎜ 8 ⎟ ⎝ c ⎠ 1 ⎛ ⎞ ln ⎜ ab ⎟ − ln ( c ) 2 8 ⎝ ⎠ 1 8 ln a + lnb − ln c 2
  • 16. Expand ⎛ a b ⎞3) ln ⎜ 8 ⎟ ⎝ c ⎠ 1 ⎛ ⎞ ln ⎜ ab ⎟ − ln ( c ) 2 8 ⎝ ⎠ 1 8 ln a + lnb − ln c 2 1 ln a + lnb − 8 ln c 2
  • 17. Combine 34) 3ln x + ln ( x + 3) 2
  • 18. Combine 34) 3ln x + ln ( x + 3) 2 3 3 ln x + ln ( x + 3) 2
  • 19. Combine 34) 3ln x + ln ( x + 3) 2 3 3 ln x + ln ( x + 3) 2 3 ⎛ 3 ⎞ ln ⎜ x ( x + 3) ⎟ 2 ⎝ ⎠
  • 20. Combine 3 4) 3ln x + ln ( x + 3) 2 3 3 ln x + ln ( x + 3) 2 3 ⎛ 3 ⎞ ln ⎜ x ( x + 3) ⎟ 2 ⎝ ⎠Book answer could be: ( ln x 3 ( x + 3) 3 )
  • 21. Combine5) log x − 3log y + log ( x + y )
  • 22. Combine 5) log x − 3log y + log ( x + y )order of operations ... from left to right
  • 23. Combine 5) log x − 3log y + log ( x + y )order of operations ... from left to right ⎛ x ⎞ log ⎜ 3 ( x + y )⎟ ⎝ y ⎠
  • 24. Combine 3 16) ln (16y ) − 2 ln x + ln y 4 4
  • 25. Combine 3 16) ln (16y ) − 2 ln x + ln y 4 4 3 1 2 ln (16y ) − ln x + ln y 4 4
  • 26. Combine 3 16) ln (16y ) − 2 ln x + ln y 4 4 3 1 2 ln (16y ) − ln x + ln y 4 4 3 1 ⎛ ⎞ 2 ln ⎜ 8y ⎟ + ln y − ln x 4 4 ⎝ ⎠
  • 27. Combine 3 16) ln (16y ) − 2 ln x + ln y 4 4 3 1 2 ln (16y ) − ln x + ln y 4 4 3 1 ⎛ ⎞ 2 ln ⎜ 8y ⎟ + ln y − ln x 4 4 ⎝ ⎠ 3 1 ⎛ ⎞ 2 ln ⎜ 8y y ⎟ − ln x 4 4 ⎝ ⎠
  • 28. Combine 3 16) ln (16y ) − 2 ln x + ln y 4 4 3 1 2 ln (16y ) − ln x + ln y 4 4 3 1 ⎛ ⎞ 2 ln ⎜ 8y ⎟ + ln y − ln x 4 4 ⎝ ⎠ 3 1 ⎛ ⎞ 2 ln ⎜ 8y y ⎟ − ln x 4 4 ⎝ ⎠ 8y ln 2 x
  • 29. Remember: Change of Base Formula
  • 30. Remember: Change of Base Formula log x ln x log b x = = logb lnb
  • 31. True or False? (Group Consensus) calculator ok to use ...
  • 32. True or False? (Group Consensus) calculator ok to use ...1) log10 ( 3 + 5 ) = log10 3 + log10 5 7 log 72) log = 2 log 23) 4 ln 6 = ln 6 4 44) 4 ln 6 = ( ln 6 )
  • 33. True or False? (Group Consensus) calculator ok to use ...1) log10 ( 3 + 5 ) = log10 3 + log10 5 False 7 log 72) log = 2 log 23) 4 ln 6 = ln 6 4 44) 4 ln 6 = ( ln 6 )
  • 34. True or False? (Group Consensus) calculator ok to use ...1) log10 ( 3 + 5 ) = log10 3 + log10 5 False 7 log 72) log = False 2 log 23) 4 ln 6 = ln 6 4 44) 4 ln 6 = ( ln 6 )
  • 35. True or False? (Group Consensus) calculator ok to use ...1) log10 ( 3 + 5 ) = log10 3 + log10 5 False 7 log 72) log = False 2 log 23) 4 ln 6 = ln 6 4 True 44) 4 ln 6 = ( ln 6 )
  • 36. True or False? (Group Consensus) calculator ok to use ...1) log10 ( 3 + 5 ) = log10 3 + log10 5 False 7 log 72) log = False 2 log 23) 4 ln 6 = ln 6 4 True 44) 4 ln 6 = ( ln 6 ) False
  • 37. HW #6I’ve got a theory that if you give 100% all of the time,somehow things will work out in the end. Larry Bird