The value of `log_((9)/(4))((1)/(2sqrt(3))sqrt(6-(1)/(2sqrt(3))sqrt(6-(1)/(2sqrt(3))sqrt(6-(1)/(2sqrt(3)))))...oo)` is

Options

A. `-2`
B. `-1`
C. `-1//2`
D. none of these

Explanation

To solve the given problem, we need to evaluate the expression:

\[ \log_{\frac{9}{4}}\left(\frac{1}{2\sqrt{3}}\sqrt{6-\frac{1}{2\sqrt{3}}\sqrt{6-\frac{1}{2\sqrt{3}}\sqrt{6-\frac{1}{2\sqrt{3}}}\ldots}}\right) \]

Let's denote the infinite nested expression as \( x \):

\[ x = \frac{1}{2\sqrt{3}}\sqrt{6 - \frac{1}{2\sqrt{3}}\sqrt{6 - \frac{1}{2\sqrt{3}}\sqrt{6 - \frac{1}{2\sqrt{3}}}\ldots}} \]

### Step 1: Set up the equation
Since the expression is nested infinitely, we can express it as:

\[ x = \frac{1}{2\sqrt{3}}\sqrt{6 - x} \]

### Step 2: Square both sides
To eliminate the square root, we square both sides:

\[ x^2 = \left(\frac{1}{2\sqrt{3}}\right)^2 (6 - x) \]

This simplifies to:

\[ x^2 = \frac{1}{12}(6 - x) \]

### Step 3: Multiply through by 12
To eliminate the fraction, we multiply both sides by 12:

\[ 12x^2 = 6 - x \]

### Step 4: Rearrange the equation
Rearranging gives us:

\[ 12x^2 + x - 6 = 0 \]

### Step 5: Factor the quadratic equation
Now we will factor the quadratic equation. We look for two numbers that multiply to \( 12 \times -6 = -72 \) and add to \( 1 \):

The factors are \( 9 \) and \( -8 \). Thus, we can write:

\[ 12x^2 + 9x - 8x - 6 = 0 \]

Grouping gives us:

\[ 3x(4x + 3) - 2(4x + 3) = 0 \]

Factoring out \( (4x + 3) \):

\[ (4x + 3)(3x - 2) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives:

1. \( 4x + 3 = 0 \) → \( x = -\frac{3}{4} \) (not valid since \( x \) must be positive)
2. \( 3x - 2 = 0 \) → \( x = \frac{2}{3} \)

### Step 7: Substitute \( x \) back into the logarithm
Now we substitute \( x \) back into the logarithm:

\[ \log_{\frac{9}{4}}\left(\frac{2}{3}\right) \]

### Step 8: Change of base formula
Using the change of base formula:

\[ \log_{\frac{9}{4}}\left(\frac{2}{3}\right) = \frac{\log\left(\frac{2}{3}\right)}{\log\left(\frac{9}{4}\right)} \]

### Step 9: Simplify the logarithm
We can simplify \( \log\left(\frac{9}{4}\right) \):

\[ \log\left(\frac{9}{4}\right) = \log(9) - \log(4) = 2\log(3) - 2\log(2) = 2(\log(3) - \log(2)) \]

Thus, we have:

\[ \log_{\frac{9}{4}}\left(\frac{2}{3}\right) = \frac{\log(2) - \log(3)}{2(\log(3) - \log(2))} \]

### Step 10: Final simplification
This simplifies to:

\[ -\frac{1}{2} \]

### Final Answer
Thus, the value of the logarithm is:

\[ \boxed{-\frac{1}{2}} \]