Exponential/ Logarithmic

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Exponential/ Logarithmic
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Exponential Functions

• f(x) ax
• Domain (-8, 8) Range (0, 8)
• Three types
• 1) if 0 lt a lt 1
• 2) if a 1
• 3) if a gt 1

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Laws of Exponents

• a x y ax ay
• ax/ ay a x y
• (ax)y axy
• (ab)x axbx

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Sketching Example

• Sketch the function y 3 2x

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Exponential Functions are One to One

• Has an inverse f-1 which is called the
  logarithmic function (loga)
• f-1(x) y f(y) x
• ay x logax y

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Example

• Find
• log10(0.001)
• log216

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Log Graph

• Reflection of exponential function about the line
  y x
• Domain (0, 8) Range (-8,8)

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Laws of Logarithms

• loga(xy) logax logay
• loga(x/y) logax logay
• logaxr rlogax

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Example

• Evaluate log280 log25

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e

• y ax
• Many formulas in calculus are greatly simplified
  if we use a base a such that the slope of the
  tangent line at y 1 is exactly 1
• For y 2x, slope at y 1 is .7
• For y 3x, slope at y 1 is 1.1
• Value of a lies between 2 and 3 and is denoted by
  the letter e
• e 2.71828

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Example

• Graph y ½ e-x 1 and find the domain and range

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Natural log (ln)

• Log with a base of e
• logex lnx
• lnx y
• ey x

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Properties of Natural Logs

• ln(ex) x
• elnx x
• ln e 1

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Example

• Find x if lnx 5

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Example

• Solve e5 3x 10

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Example

• Express ln a ½ ln b as a single logarithm

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Expression

• y logax
• ay x
• ln ay ln x
• y ln a ln x
• y ln x/ ln a
• logax ln x/ ln a if a ? 0

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Example

• Evaluate log85

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Example

• The half-life of a radioactive substance given by
  f(t) 24 2-t/25 Find the inverse