How do I find the inverse of this exponential function?

Question

$x=-3(3^{-x})+9$

I know the steps up until a certain point.

$x=-3(3^{-y})+9$

$x-9=-3(3^{-y})$

$\frac{(x-9)}{-3} = 3^y$

$ln (\frac{x-9}{-3}) = -y * ln 3$

Not sure what to do from here. I know I have to get y by itself but thats it. Can anyone help please?

Answer 1

We want to solve for $y$, after switching $x$ and $y$.

$y=-3(3^{-x})+9$

becomes

$x=-3(3^{-y})+9\\ \implies x-9=-3(3^{-y})\\ \implies \dfrac{x-9}{-3}=3^{-y}\\ \implies -\dfrac{1}{3}x+3=3^{-y}\\ \implies \ln\left(-\dfrac{1}{3}x+3\right)=\ln (3^{-y})\\ \implies \ln\left(-\dfrac{1}{3}x+3\right)=-y\cdot \ln 3\\ \implies -\dfrac{ \ln\left(-\dfrac{1}{3}x+3\right)}{\ln 3}=y$

Answer 2

Divide both sides by $-\ln 3$ and you are there