Find the solution of the inequality `2log_(1/4)(x+5)>9/4log_(1/(3sqrt(3)))(9)+log_(sqrt(x+5))(2)`

Options

A. `(-5,-4)`
B. `(-3,-1)`
C. `(-4,-1)`
D. `(-5,-2)`

Explanation

To solve the inequality \( 2\log_{1/4}(x+5) > \frac{9}{4}\log_{1/(3\sqrt{3})}(9) + \log_{\sqrt{x+5}}(2) \), we will follow these steps:

### Step 1: Simplify the logarithmic terms
We start by simplifying each logarithmic term in the inequality.

1. **First term**:
\[ 2\log_{1/4}(x+5) = 2\log_{2^{-2}}(x+5) = 2 \cdot \frac{-\log_2(x+5)}{-2} = -\log_2(x+5) \]

2. **Second term**:
\[ \frac{9}{4}\log_{1/(3\sqrt{3})}(9) = \frac{9}{4} \cdot \frac{-\log_3(9)}{-\log_3(3\sqrt{3})} \]
Since \(\log_3(9) = 2\) and \(\log_3(3\sqrt{3}) = 1 + \frac{1}{2} = \frac{3}{2}\),
\[ = \frac{9}{4} \cdot \frac{-2}{-\frac{3}{2}} = \frac{9}{4} \cdot \frac{4}{3} = 3 \]

3. **Third term**:
\[ \log_{\sqrt{x+5}}(2) = \frac{\log_2(2)}{\log_2(\sqrt{x+5})} = \frac{1}{\frac{1}{2}\log_2(x+5)} = \frac{2}{\log_2(x+5)} \]

### Step 2: Substitute back into the inequality
Now we substitute the simplified terms back into the inequality:
\[ -\log_2(x+5) > 3 + \frac{2}{\log_2(x+5)} \]

### Step 3: Let \( y = \log_2(x+5) \)
Substituting \( y \) into the inequality gives:
\[ -y > 3 + \frac{2}{y} \]
Multiplying through by -1 (and reversing the inequality):
\[ y < -3 - \frac{2}{y} \]

### Step 4: Multiply through by \( y \) (assuming \( y \neq 0 \))
\[ y^2 + 3y + 2 < 0 \]

### Step 5: Factor the quadratic
Factoring gives:
\[ (y + 1)(y + 2) < 0 \]

### Step 6: Determine the intervals
The roots of the equation are \( y = -1 \) and \( y = -2 \). The quadratic opens upwards, so the solution to the inequality is:
\[ -2 < y < -1 \]

### Step 7: Substitute back for \( y \)
Substituting back for \( y \):
\[ -2 < \log_2(x+5) < -1 \]

### Step 8: Convert back to exponential form
This gives us:
\[ 2^{-2} < x + 5 < 2^{-1} \]
Which simplifies to:
\[ \frac{1}{4} < x + 5 < \frac{1}{2} \]

### Step 9: Solve for \( x \)
Subtracting 5 from all parts:
\[ \frac{1}{4} - 5 < x < \frac{1}{2} - 5 \]
This simplifies to:
\[ -\frac{19}{4} < x < -\frac{9}{2} \]

### Final Answer
Thus, the solution to the inequality is:
\[ x \in \left(-\frac{19}{4}, -\frac{9}{2}\right) \]