To solve this equation, we need to first rewrite both sides of the equation in terms of the same base.

We know that 5=5^1, so we can rewrite the equation as:
(2/9)^(2x+3) = 5^(x-2) = 5^(1)*(x-2).

Next, we can rewrite 5 as 5^(1), and apply the power rule for exponents:
(2/9)^(2x+3) = 5^(1)(x-2)
(2/9)^(2x+3) = 5^(x-2) = 5^(x)5^(-2).

Now, we have both sides of the equation written in terms of the same base. We can rewrite the equation as:
(2/9)^(2x+3) = 5^(x)*5^(-2).

Next, apply the properties of exponents to simplify the equation:
(2/9)^(2x+3) = 5^(x)5^(-2)
(2/9)^(2x+3) = 5^(x)1/25
(2/9)^(2x+3) = 5^(x)/25.

Now, we can rewrite the equation as:
(2/9)^(2x+3) = 5^(x)/25.

To solve for x, we need to compare the exponents on both sides of the equation. This can be done by taking the natural logarithm of both sides to eliminate the exponents.

ln((2/9)^(2x+3)) = ln(5^(x)/25).

Next, apply the power rule of logarithms:
(2x+3)ln(2/9) = xln(5) - ln(25).

Now, isolate x by distributing and rearranging the terms:
2xln(2/9) + 3ln(2/9) = xln(5) - ln(25)
2xln(2/9) - xln(5) = -3ln(2/9) - ln(25)
x(2ln(2/9) - ln(5)) = -3ln(2/9) - ln(25)
x = (-3ln(2/9) - ln(25)) / (2ln(2/9) - ln(5))

This is the solution for x in terms of natural logarithms.