log9 (3log2 (1 + log3 (1 + 2log2x))) = 1/2. Find x.

Sure, let's solve this step by step:

1. Given equation is: log9 (3log2 (1 + log3 (1 + 2log2x))) = 1/2

2. We know that loga(b) = c implies a^c = b. So, we can rewrite the equation as: 9^(1/2) = 3log2 (1 + log3 (1 + 2log2x))

3. Simplifying 9^(1/2) gives us 3. So, the equation becomes: 3 = 3log2 (1 + log3 (1 + 2log2x))

4. Divide both sides by 3 to get: 1 = log2 (1 + log3 (1 + 2log2x))

5. Using the property of logarithms, we can rewrite the equation as: 2^1 = 1 + log3 (1 + 2log2x)

6. Simplifying 2^1 gives us 2. So, the equation becomes: 2 = 1 + log3 (1 + 2log2x)

7. Subtract 1 from both sides to get: 1 = log3 (1 + 2log2x)

8. Using the property of logarithms, we can rewrite the equation as: 3^1 = 1 + 2log2x

9. Simplifying 3^1 gives us 3. So, the equation becomes: 3 = 1 + 2log2x

10. Subtract 1 from both sides to get: 2 = 2log2x

11. Divide both sides by 2 to get: 1 = log2x

12. Using the property of logarithms, we can rewrite the equation as: 2^1 = x

13. Simplifying 2^1 gives us 2. So, the solution is x = 2.