# Thread: Inverse Function Problem (involving logarithms)

1. ## 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

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

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.

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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.

3. Thank you so much. I feel stupid for not seeing what was so clear. Sometimes all it takes is a little outside help.