Chapter 5: Exponential and Logarithmic Functions 5.5: Properties and Laws of Logarithms

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Chapter 5: Exponential and Logarithmic Functions 5.5: Properties and Laws of Logarithms. Essential Question: What are the three properties that simplify logarithmic expressions? Describe how to use them. 5.5: Properties and Laws of Logarithms. Basic Properties of Logarithms - PowerPoint PPT Presentation

Transcript of Chapter 5: Exponential and Logarithmic Functions 5.5: Properties and Laws of Logarithms

  • Chapter 5: Exponential and Logarithmic Functions5.5: Properties and Laws of LogarithmsEssential Question: What are the three properties that simplify logarithmic expressions? Describe how to use them.

  • 5.5: Properties and Laws of LogarithmsBasic Properties of LogarithmsLogarithms are only defined for positive real numbersNot possible for 10 or e to be taken to an exponent and result in a negative numberLog 1 = 0 and ln 1 = 0100 = 1 & e0 = 1Log 10k = k and ln ek = klog10104 = k10k = 104k = 410log v = v and eln v = v10log 22 = vlog10v = log 22v = 22

  • 5.5: Properties and Laws of LogarithmsSolving Equations by Using Properties of Logarithmsln(x + 1) = 2Method #1e2 = x + 1e2 1 = xx 6.3891Method #2eln(x + 1) = e2x + 1 = e2See method #1 above

  • 5.5: Properties and Laws of LogarithmsProduct Law of LogarithmsLaw of exponents states bmbn = bm+nBecause logarithms are exponents:log (vw) = log v + log wln (vw) = ln v + ln wProof:vw = 10log v 10log w = 10log v + log wvw = 10log vwTaking from above:10log v + log w = 10log vwlog v + log w = log vwProof of ln/e works the same way

  • 5.5: Properties and Laws of LogarithmsProduct Law of Logarithms (Application)Given that log 3 = 0.4771 and log 11 = 1.0414 find log 33log 33= log (3 11) = log 3 + log 11 = 0.4771 + 1.0414 = 1.5185Given that ln 7 = 1.9459 and ln 9 = 2.1972 find ln 63ln 63= ln (7 9) = ln 7 + ln 9 = 1.9459 + 2.1972 = 4.1431

  • 5.5: Properties and Laws of LogarithmsQuotient Law of LogarithmsLaw of exponents states Because logarithms are exponents:log ( ) = log v log w ln ( ) = ln v ln w Proof is the same as the Product Law

  • 5.5: Properties and Laws of LogarithmsQuotient Law of Logarithms (Application)Given that log 28 = 1.4472 and log 7 = 0.8451 find log 4log 4= log (28 / 7) = log 28 log 7 = 1.4472 0.8451 = 0.6021Given that ln 18 = 2.8904 and ln 6 = 1.7918 find ln 3ln 3= ln (18 / 6) = ln 18 ln 6 = 2.8904 1.7918 = 1.0986

  • 5.5: Properties and Laws of LogarithmsPower Law of LogarithmsLaw of exponents states (bm)k = bmkBecause logarithms are exponents:log (vk) = k log vln (vk) = k ln vProof:v = 10log v vk = (10log v)k = 10k log vvk = 10log vkTaking from above:10k log v = 10log vkk log v = log vkProof of ln/e works the same way

  • 5.5: Properties and Laws of LogarithmsPower Law of Logarithms (Application)Given that log 6 = 0.7782 find log log = log 6 = log 6 = (0.7782) = 0.3891Given that ln 50 = 3.9120 find ln

  • 5.5: Properties and Laws of LogarithmsSimplifying ExpressionsWrite as a single logarithm:ln 3x + 4 ln x ln 3xy

  • 5.5: Properties and Laws of LogarithmsSimplifying ExpressionsWrite as a single logarithm:

  • 5.5: Properties and Laws of LogarithmsAssignmentPage 369Problems 1-25, odd problemsShow work