log9 (3log2 (1 + log3 (1 + 2log2x))) = 1/2. Find x.Choices:- None of the above 1 12/30 2

To solve the equation, we need to follow these steps:

1. We know that log base a of a equals 1. So, we can rewrite the equation as: 3log2 (1 + log3 (1 + 2log2x)) = 9^(1/2) = 3.

2. Divide both sides by 3 to isolate the inner expression: log2 (1 + log3 (1 + 2log2x)) = 1.

3. Use the property of logarithms that says log base a of a equals 1 to rewrite the equation as: 1 + log3 (1 + 2log2x) = 2^1 = 2.

4. Subtract 1 from both sides to isolate the inner expression: log3 (1 + 2log2x) = 2 - 1 = 1.

5. Use the property of logarithms that says log base a of a equals 1 to rewrite the equation as: 1 + 2log2x = 3^1 = 3.

6. Subtract 1 from both sides to isolate the inner expression: 2log2x = 3 - 1 = 2.

7. Divide both sides by 2 to isolate the inner expression: log2x = 2 / 2 = 1.

8. Use the property of logarithms that says log base a of a equals 1 to rewrite the equation as: x = 2^1 = 2.

So, the solution to the equation is x = 2.

To solve the equation, we need to follow these steps:

1. We start with the given equation: log9 (3log2 (1 + log3 (1 + 2log2x))) = 1/2.

2. We know that log base b of a equals c can be written as b^c = a. 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. Dividing both sides by 3, we 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. Subtracting 1 from both sides, we get: 1 = log3 (1 + 2log2x).

8. Using the property of logarithms again, 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. Subtracting 1 from both sides, we get: 2 = 2log2x.

11. Dividing both sides by 2, we get: 1 = log2x.

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

13. Simplifying 2^1 gives us 2, so the equation becomes: x = 2.

So, the answer is 2.