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66 Class Note for MATH 140 at PSU

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 32 views.

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Date Created: 02/06/15
Laws of Logarithms There are very few laws of logarithms that let us work with them very effectively despite the fact that logarithms are very hard to evaluate in general Assume a gt 0 is a positive real number 0 logaz y means I ay for real numbers I gt 0 and y So 0 logaax z for every real number 0 034 z for every I gt 0 The combination of the last two statements says that logs and exponential functions are inverse functions However you should be careful to keep track of when I must be strictly positive since no logarithm can be de ned at zero and the logarithm of a negative number is a complex number which we also won7t deal with The laws of exponents lead to the following laws of logarithmsi Here we assume I and y are positive real numbers 1 loamy 10ml 10gay 2 loga 10ml 7 10gay 3 log 17 Tlogaz for any real number Ti We will generally only care about the case when a e in which case loga lnz is the natural logarithmi One excuse for only working with this most important logarithm is the following change of base formula77 for a f l Inltzgt loga 7 may For the natural log the laws become 1 lnzy lnz lny 2 ln lnz 7 lny 3i lnzT Tlnz for any real number Ti Also the earlier statements become 0 lnz y means I ey for real numbers I gt 0 and y o lne z for every real number 0 elm z for every I gt 0 One way we can use the laws is to collapse some large expression involving several logs Example log3 z 5 log3 z 7 5 7 4 log32 log312 7 25 7 log316 12 7 25 logs T Another that Will be very important later is really the opposite idea We take the natural log of a complicated expression and break it down into a number of manageable piecesi Example 156I2 47 in1 5 nI27 77nI37 8 lnlt IMP 71lt5gt1lt 4 1lt s 6lnz 5 7lnz2 7 4 7 Slnz3 7 5

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