Logarithms with Variables

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Remember that

logb y = x

can be defined as an exponential expression like this:

y = bx

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

  • log2 y = x
    y = 2x
  • 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:

logb (n + 1) = x

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

(n + 1) = bx

  • log2 (n + 1) = x
    n + 1 = 2x
  • 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 = 2x
    log2 8 = x = 3
  • 27 = 3x
    log3 27 = x = 3
  • 25 = 5x
    log5 25 = x = 2
  • 36 = 6x
    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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