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Inverse Exponential Functions

  1. Aug 23, 2007 #1
    First of all, sorry for asking so many questions. I do not want answers, just a method of solving them.

    1. The problem statement, all variables and given/known data

    For the exponential function f(x) = ab^x, suppose f(2) = 2 and f(4) = 18.

    a. Find a and b.
    b. Find f^-1(54), the inverse function.

    2. Relevant equations

    None, really...

    3. The attempt at a solution

    a. f(2) = 2, meaning ab^2 = 2. f(4) = 18, meaning ab^4 = 18.

    (ab^2 = 2) * 9 -> 9ab^2 = 18
    ab^4 = 18

    9ab^2 = ab^4

    divide both sides by ab^2

    9 = b^2
    b = 3

    a(3)^2 = 2
    9a = 2
    a = 2/9

    b. The inverse of ab^54...

    Inverse of: (2/9)(3)^54

    And from here, I'm lost. Do you fifty-fourth root everything because it is the inverse?
     
  2. jcsd
  3. Aug 23, 2007 #2

    Doc Al

    User Avatar

    Staff: Mentor

    inverse

    This means: find the value of x such that ab^x = 54.
     
  4. Aug 23, 2007 #3
    that's what the inverse does to the function?
     
  5. Aug 23, 2007 #4
    that gives me 3^x = 243. using logarithms...

    x log3 = log 243

    x = log 243/ log 3

    x = 5

    am i correct?
     
  6. Aug 23, 2007 #5

    Doc Al

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    Staff: Mentor

    Plug your answer into the original equation and see for yourself! :smile:
     
  7. Aug 23, 2007 #6
    3^5 is indeed 243.

    Thanks, Doc Al.
     
  8. Aug 28, 2007 #7
    One thing I remember about the inverse function is that it's the function that for a value of y, would return the x-value that the original function used.

    f(x)
    x | y
    -----
    1 | 3
    2 | 12
    3 | 27
    4 | 48
    5 | 75

    So for
    f^-1(x)
    x | y
    -----
    3 | 1
    12 | 2
    27 | 3
    48 | 4
    75 | 5

    So:

    y = ab^x
    Solve for x to get the inverse function:

    x = lnb / (lny - lna)

    I think. I'm not sure at all.
     
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