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Exploring Logarithmic Functions: Understanding Logs and Their Properties

Dive into the world of logarithmic functions to understand logs, their properties, and why they are essential in mathematics. Explore how logarithms work, their applications, and graphing techniques with relevant examples.

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Exploring Logarithmic Functions: Understanding Logs and Their Properties

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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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