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  • 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. Important Idea The definitions of common and natural logarithms differ only in their bases, therefore, they share the same properties and laws.
  • 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. 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. Example Use the properties of logarithms to solve the equation: ln( 1) 2x e e+ = log( 2) 2x − = ln( 4) 2x + = −
  • 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. 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. Example log33Find given log3 .4771= and log11 1.0414= What law was used?
  • 9. Example ln63Find given ln7 1.9459= and ln9 2.1972= What law was used?
  • 10. Try This find given and log12 log6 .7782= log2 .3010= log12 1.0792= Using the product law,
  • 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. 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. 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. Example log3Find given log12 1.0792= and log4 .6021= What law was used?
  • 15. Try This log3Find given log6 .7782= and log2 .3010= log3 .4771=
  • 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. 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. 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. 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. 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. 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. Example Use a combination of logarithmic properties and laws to re-write the given expression: 2 2( 3) ln 1 x x  +  ÷ − 
  • 23. Example Use a combination of logarithmic properties and laws to re-write the given expression: 3 10 log 1 x x    ÷ + 
  • 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. 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. 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. 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.