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Math Help - Inverse Function Problem (involving logarithms)

  1. #1
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    Inverse Function Problem (involving logarithms)

    Hello, this problem is actually for a Calculus II course, but I feel the problem I'm having is figuring out the algebraic complications. The problem is as follows:

    Find the inverse of f(x) if f(x) = (-6-3*3^x)/(-5-7*3^x).

    Now, I understand that to find the inverse of a function, the first step would be to switch the x and y variables, which I preceeded to do. Next, I took the natural log of both sides and got:

    ln(x) = ln(-6-3*3^(y)) - ln(-5-7*3^(3))

    This is the where I'm unsure of what to do. First, I thought I might be able to take natural number e to both sides. Is this a mistake? I may have the wrong approach altogether. If you have any hints or tips as to how I should approach this problem, aside from the aforementioned routes, I would greatly appreciate it.

    Sincerely,

    Austin Martin
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  2. #2
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    f(x) = \frac{-6 - 3\cdot 3^{x}}{-5-7\cdot 3^{x}}

    Let y = f^{-1}(x) (just for visual and latex-ing purposes):

    x = \frac{-6 - 3 \cdot 3^{y}}{-5 - 7\cdot 3^{y}}
    x\left(-5 - 7\cdot 3^{y}\right) = -6 - 3\cdot 3^{y}............... (Got rid of the denominator)
    -5x - (7x)(3^{y}) = -6 - (3)(3^{y}) ...........(Distribute x)
    (3)(3^y) - (7x)(3^y) = -6 + 5x............. (Collect terms containing 3^y)
    3^{y} \left(3 - 7x\right) = -6 + 5x....................... (Factor 3^y out)

    Divide both sides by (3 - 7x) to isolate 3^{y}. Now you can apply the logarithmic function to both sides to isolate y.

    ------

    This is the general format of these questions: Isolate the term containing your variable in the exponent, take the logarithm of both sides and voila.
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  3. #3
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    Thank you so much. I feel stupid for not seeing what was so clear. Sometimes all it takes is a little outside help.
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