To find the value of 1000x, we need to solve the given equation and substitute the result back into the expression.

Let's break down the problem and solve it step by step:

Step 1: Solve the given equation
The equation is log(x * log(√2 * √8)) = 1/3.

Step 2: Simplify the equation
Using the properties of logarithms, we can simplify the equation further.
Since log(a * b) = log(a) + log(b), we can rewrite the equation as:
log(x) + log(log(√2 * √8)) = 1/3.

Step 3: Simplify the expression inside the logarithm
We know that √2 * √8 = √(2 * 8) = √16 = 4.
So, log(x) + log(log(4)) = 1/3.

Step 4: Simplify the logarithm of 4
Since log(4) = log(2^2) = 2 * log(2), we can rewrite the equation as:
log(x) + log(2 * log(2)) = 1/3.

Step 5: Combine the logarithms
Using the property log(a) + log(b) = log(a * b), we can simplify the equation further:
log(x * 2 * log(2)) = 1/3.

Step 6: Convert the equation into exponential form
Since log(a) = b is equivalent to a = 10^b, we can rewrite the equation as:
x * 2 * log(2) = 10^(1/3).

Step 7: Solve for x
Divide both sides of the equation by 2 * log(2):
x = 10^(1/3) / (2 * log(2)).

Step 8: Calculate the value of x
Using a calculator, we can evaluate the right side of the equation to find the value of x.

Step 9: Substitute the value of x into the expression
Now that we have the value of x, we can substitute it back into the expression 1000x.

Step 10: Calculate the value of 1000x
Multiply the value of x by 1000 to find the final result.