You made a mistake with step (3) - you can't just raise the ln functions to the power of e; you must do the same to both sides of the equation:

e^(r(x)) = e^(2 ln 1.8 + x ln1.8) = e^(2ln1.8) * e^(x ln 1.8)

I think it's easier to find the inverse like this:

r = 2 ln (1.8 * 1.8^x) = 2 ln (1.8)^(x+1) = 2(x+1) ln(1.8)

Divide through by 2 ln(1.8): r/(2 ln (1.8)) = x+1

Rearrange to get x by itself: x= r/(2 ln(1.8)) - 1

So if r = 9.4 you have x = 9.4/(2 ln(1.8)) - 1 = 6.996

To check that this is correct, try x=6.996 in the original equation: r = 2 ln(1.8 * 1.8^6.996) = 2 ln(1.8^7.996) = 2 ln(109.9) = 9.4