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Sections 5.3-5.5

Sections 5.3-5.5. Logarithmic Functions (5.3). What is a logarithm??? LOGS ARE POWERS!!!!

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Sections 5.3-5.5

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  1. Sections 5.3-5.5

  2. Logarithmic Functions (5.3) • What is a logarithm??? • LOGS ARE POWERS!!!! • A logarithm or “log” of a number of a certain base is the exponent to which the base of the log must be raised in order to produce the number. The base cannot equal 1 and must be greater than 0. • For instance, if logb(x) = c and b≠1 andb>0, then c is the specific exponent to which you must raise b in order to get x: bc=x

  3. Logarithmic Functions • Why do we need logs?Let’s explore… 32 = 9 and 33 = 27 but what would we need to raise 3 to in order to get 20?? 3a = 20 that’s what logs tell us!! a = log320 Which two integers is log320 between? 2 and 3

  4. Logarithmic Functions • From the definition, we have stated that if logb(x) = c, then bc=x under the conditions that b≠1 andb>0. • Why do we need to place any restrictions on b or x so this can make sense? Let’s try some values…

  5. logb(x) = c, sobc=x log2(8) = c so2c = 8 c = 3, so far we are ok log1(5) = c so1c=5 Does not exist; 1c always equals 1 log-2(8) = c so(-2)c=8 Does not exist; if c = 3, then (-2)3 = -8 log3(-9) = c so3c=-9 Does not exist; 3c cannot be negative log2(0) = c so2c=0 Does not exist; 2c cannot equal 0 Summary: b≠1, b>0 and x>0

  6. Log Properties • logbb = 1 • logb1 = 0 • common log has base 10: log(x) = log10(x) • natural log has base e: ln(x) = loge(x) • Therefore… • log10 = 1 • lne = 1

  7. Practice • Evaluate, if possible. If not, state so.

  8. y = logb(x - h) + k • When graphing logs we first need to identify and graph the asymptote. Earlier we discovered that the argument inside the log must be greater than 0. • Therefore, x > h so the domain is (h, +∞) and there must be an asymptote at x = h • The range is all real numbers • Now find three points; the simplest values are when x - h = 1 and when x - h = b

  9. Graph of a Logarithmic Function • Graph of y = logb(x) when 0<b<1 • Graph of y = logb(x) when b>1 • Can you state any characteristics? • Asymptotes, x - intercepts, Domain, Range

  10. Practice • Graph. State the domain and range.

  11. Change of Base Formula This formula allows us to compute logs using the calculator, by converting to base 10 or e. Example:

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