Logarithms with Variables

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Remember that

log_b y = x

can be defined as an exponential expression like this:

y = b^x

Many exercises expect you to convert from logarithmic to exponential expression as in these flash cards:

• log₂ y = x
  y = 2^x
• log₂ y = 3
  y = 2³
• log₃ y = 4
  y = 3⁴
• log₃ y = 7
  y = 3⁷
• log₄ y = 3
  y = 4³
• log₄ y = 5
  y = 4⁵

Many exercises introduce a variable into the y value like this:

log_b (n + 1) = x

You can easily solve for n by converting to an exponential expression like this:

(n + 1) = b^x

• log₂ (n + 1) = x
  n + 1 = 2^x
• log₂ (n + 1) = 3
  n + 1 = 2³
• log₃ (n + 2) = 4
  n + 2 = 3⁴
• log₃ (n + 4) = 5
  n + 4 = 3⁵
• log₄ (n + 6) = 3
  n + 6 = 4³
• log₄ (n + 7) = 5
  n + 7 = 4⁵

And if you are really BOLD you will use logarithms to solve for exponents:

• 8 = 2^x
  log₂ 8 = x = 3
• 27 = 3^x
  log₃ 27 = x = 3
• 25 = 5^x
  log₅ 25 = x = 2
• 36 = 6^x
  log₆ 36 = x = 2
• 500 = e^.05t
  ln 500 = .05t
• 600 = 300e^.07t
  ln 2 = .07t
• 1500 = 500e^.08t
  ln 3 = .08t
• 2500 = 1000e^.09t
  ln 2.5 = .09t

Be sure to contact Fred if you would like further explanations of these flash cards.

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