To solve the equation, we can rewrite 2^(8/3) and 3^4 as a common base.

2^(8/3) = (2^2)^(8/6) = 4^(8/6) = 4^(4/3)

3^4 = (3^(4/3))^3 = 9^(4/3)

Now, the equation becomes:

(3/4)^x = 4^(4/3) / 9^(4/3)

To simplify further, we can rewrite 4 and 9 as powers of 2 and 3 respectively.

4^(4/3) = (2^2)^(4/3) = 2^(8/3)

9^(4/3) = (3^2)^(4/3) = 3^(8/3)

Now, the equation becomes:

(3/4)^x = 2^(8/3) / 3^(8/3)

To simplify further, we can rewrite 3/4 as (3^1)/(2^2).

((3^1)/(2^2))^x = 2^(8/3) / 3^(8/3)

Now, we can apply the exponent rule (a^m / a^n = a^(m-n)) to simplify the right side of the equation.

((3^1)/(2^2))^x = 2^(8/3 - 8/3)

((3^1)/(2^2))^x = 2^0

((3^1)/(2^2))^x = 1

Now, we can simplify the left side of the equation.

((3^1)/(2^2))^x = (3^x)/(2^(2x))

Therefore, the equation becomes:

(3^x)/(2^(2x)) = 1

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

log((3^x)/(2^(2x))) = log(1)

Using the logarithm properties, we can simplify the equation further.

log(3^x) - log(2^(2x)) = 0

x * log(3) - 2x * log(2) = 0

Now, we can factor out x.

x * (log(3) - 2 * log(2)) = 0

Since the product of two factors is zero if and only if at least one of the factors is zero, we have two cases to consider:

Case 1: x = 0

Case 2: log(3) - 2 * log(2) = 0

To solve for x in Case 2, we can isolate log(3) by moving -2 * log(2) to the other side of the equation.

log(3) = 2 * log(2)

Now, we can rewrite the equation in exponential form.

3 = 2^2

Since 3 is not equal to 4, there is no solution for x in Case 2.

Therefore, the only solution to the equation is x = 0.