Hprec5 5

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Hprec5 5

  1. 1. 5.5: Properties & Laws of Logarithms © 2008 Roy L. Gover(www.mrgover.com) Learning Goals: •Use properties and laws of logarithms to simplify and evaluate expressions.
  2. 2. Important Idea The definitions of common and natural logarithms differ only in their bases, therefore, they share the same properties and laws.
  3. 3. Important Idea Properties of Common Logarithms: •log v defined only for v>0 •log 1=0 & log 10=1 • log10k k= • for v>0log 10 v v=
  4. 4. Important Idea Properties of Natural Logarithms: •ln v defined only for v>0 •ln 1=0 & ln e=1 • ln k e k= • for v>0 lnv e v=
  5. 5. Example Use the properties of logarithms to solve the equation: ln( 1) 2x e e+ = log( 2) 2x − = ln( 4) 2x + = −
  6. 6. Try this Use the properties of logarithms to solve the equation: log( 3) 1x − = x=13 3 5.718x e= + ≈ln( 3) 1x − =
  7. 7. Important Idea ln(ab)= ln a + ln b ln an =n ln a ln ln ln a a b b   = −    Product Law: Quotient Law: Power Law:
  8. 8. Example log33Find given log3 .4771= and log11 1.0414= What law was used?
  9. 9. Example ln63Find given ln7 1.9459= and ln9 2.1972= What law was used?
  10. 10. Try This find given and log12 log6 .7782= log2 .3010= log12 1.0792= Using the product law,
  11. 11. Example Using the product law, write the given expression as a single logarithm: 2 ln lnx x+ log(2 ) log( 1)x x+ +
  12. 12. Try This Using the product law, write the given expression as a single logarithm: ln( 1) ln( 1)x x+ + − 2 ln( 1)x −
  13. 13. Try This Using the product law, write the given expression as two logarithms: 2 ln( 2)x x+ − ln( 2) ln( 1)x x+ + −
  14. 14. Example log3Find given log12 1.0792= and log4 .6021= What law was used?
  15. 15. Try This log3Find given log6 .7782= and log2 .3010= log3 .4771=
  16. 16. Example Using the quotient law, write the given expression as a single logarithm: 2 ln lnx x− log(2 ) log( 1)x x− +
  17. 17. Try This Using the quotient law, write the given expression as a single logarithm: ln( 1) ln( 1)x x+ − − 1 ln 1 x x +   ÷ − 
  18. 18. Try This Using the quotient law, write the given expression as two logarithms: ln( 1) ln( 1)x x+ − − 1 ln 1 x x +   ÷ − 
  19. 19. Example Using the power law, re- write the given expression and simplify if possible: 2 ln x 3log(2 )x ( 1) ln( 1) x x + + ( 1)ln( 1)x x+ +
  20. 20. Example Using the power law, re- write the given expression and simplify if possible: 2 ln x 3log(2 )x ( 1) ln( 1) x x + + ( 1)ln( 1)x x+ +
  21. 21. Try This Using the power law, re- write the given expression and simplify if possible: 2 log4 3 lne 5 log10 2log4 3ln 3e = 5log10 5=
  22. 22. Example Use a combination of logarithmic properties and laws to re-write the given expression: 2 2( 3) ln 1 x x  +  ÷ − 
  23. 23. Example Use a combination of logarithmic properties and laws to re-write the given expression: 3 10 log 1 x x    ÷ + 
  24. 24. Try This Use a combination of logarithmic properties and laws to re- write the given expression: 3 ( 5) ln 1 e x x  −  ÷ +  1 3ln( 5) ln( 1)x x+ − − +
  25. 25. Example The 1989 world series earthquake in San Francisco measure 7.0 on the Richter Scale. The great earthquake of 1906 measured 8.3. How much more intense was the 1906 quake? ( )0logR i i=
  26. 26. Example Decibels are calculated by the function where is the minimum sound intensity detectable by the human ear. Find the decibel level of a jet engine which is 10 billion times 010log( )i i 0i 0.i
  27. 27. Lesson Close The manipulation of logarithms is a fundamental math skill that you will need in upper level math courses and in science and engineering.

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