To solve this equation, we can simplify it by using properties of logarithms:

We know that log(a^b) = b * log(a) and log(a) - log(b) = log(a/b)

So, we can rewrite the equation as:

lg^(-1)x + 8lg(x) + 9 = 0

Now, let's rewrite lg^(-1)x as 1/lg(x):

1/lg(x) + 8lg(x) + 9 = 0

Now, let's combine the terms and simplify:

8lg(x) - 1/lg(x) = -9

Now we have a quadratic equation in terms of lg(x):

8(lg(x))^2 - 1 = -9(lg(x))

8(lg(x))^2 + 9(lg(x)) - 1 = 0

Let's substitute u = lg(x):

8u^2 + 9u - 1 = 0

Now, we can solve this quadratic equation for u using the quadratic formula:

u = (-9±√(9^2 - 48(-1)))/(2*8)
u = (-9±√(81 + 32))/16
u = (-9±√113)/16

Now, we substitute back u = lg(x):

lg(x) = (-9±√113)/16

Now, we can solve for x:

x = 10^((-9±√113)/16)