If `(3sqrt4)^(2x+1/2)=1/32`, then x =

Options

A. -2
B. 4
C. -6
D. -4

Answer

To solve the equation \((3\sqrt{4})^{(2x + \frac{1}{2})} = \frac{1}{32}\), we will follow these steps:

### Step 1: Simplify the left side
First, we can rewrite \(3\sqrt{4}\):
\[ 3\sqrt{4} = 3 \cdot 2 = 6 \]
So, the equation becomes:
\[ 6^{(2x + \frac{1}{2})} = \frac{1}{32} \]

### Step 2: Rewrite \(\frac{1}{32}\) as a power of 2
We know that \(32 = 2^5\), therefore:
\[ \frac{1}{32} = 2^{-5} \]
Now, we need to express \(6\) in terms of base \(2\). However, we can also express \(32\) in terms of base \(6\) by recognizing that \(6\) is not a power of \(2\). Instead, we can directly solve for \(x\) by taking logarithms.

### Step 3: Take logarithm of both sides
Taking logarithm base \(6\) on both sides:
\[ 2x + \frac{1}{2} = \log_6\left(\frac{1}{32}\right) \]

### Step 4: Simplify the logarithm
Using the change of base formula:
\[ \log_6\left(\frac{1}{32}\right) = \log_6(2^{-5}) = -5 \cdot \log_6(2) \]
Now we have:
\[ 2x + \frac{1}{2} = -5 \cdot \log_6(2) \]

### Step 5: Solve for \(2x\)
Rearranging gives:
\[ 2x = -5 \cdot \log_6(2) - \frac{1}{2} \]

### Step 6: Divide by 2 to find \(x\)
Now, divide everything by \(2\):
\[ x = \frac{-5 \cdot \log_6(2) - \frac{1}{2}}{2} \]

### Step 7: Final simplification
This gives:
\[ x = -\frac{5}{2} \cdot \log_6(2) - \frac{1}{4} \]

### Conclusion
The value of \(x\) can be calculated using the logarithm value, but it can also be approximated or left in this form depending on the context of the question.