To solve the equation, we can rewrite it as:

9 * (5x/3)^x + 4 = 1/243

First, let's simplify the equation by simplifying the exponent:

(5x/3)^x = (5^x * x^x) / (3^x)

Now, substitute this back into the equation:

9 * ((5^x * x^x) / (3^x)) + 4 = 1/243

To simplify further, let's multiply both sides of the equation by 243 to eliminate the fraction:

243 * 9 * ((5^x * x^x) / (3^x)) + 243 * 4 = 1

Simplifying:

2187 * (5^x * x^x) / (3^x) + 972 = 1

Now, let's isolate the term with the variable:

2187 * (5^x * x^x) / (3^x) = 1 - 972

2187 * (5^x * x^x) / (3^x) = -971

To simplify further, let's multiply both sides of the equation by (3^x):

2187 * (5^x * x^x) = -971 * (3^x)

Now, let's divide both sides of the equation by -971:

(2187 * (5^x * x^x)) / -971 = 3^x

To solve for x, we can take the logarithm of both sides of the equation:

log((2187 * (5^x * x^x)) / -971) = log(3^x)

Using logarithm properties, we can simplify further:

log(2187) + log(5^x * x^x) - log(971) = x * log(3)

Now, let's isolate the x term:

x * log(3) - log(5^x * x^x) = log(2187) - log(971)

Finally, divide both sides of the equation by log(3) to solve for x:

x = (log(2187) - log(971)) / (log(3) - log(5^x * x^x))

This is the general solution for x in the equation. However, it may not have a simple numerical solution without further simplification or approximation.