The value of `log_(3sqrt(2)) 5832` is:

Options

A. `1/4`
B. `-5`
C. `-4`
D. `-3`

Answer

To find the value of \( \log_{3\sqrt{2}} 5832 \), we can follow these steps:

### Step 1: Factor 5832
First, we need to factor the number 5832 to express it in terms of its prime factors.

**Calculation:**
- Divide 5832 by 2:
\( 5832 \div 2 = 2916 \)
- Divide 2916 by 2:
\( 2916 \div 2 = 1458 \)
- Divide 1458 by 2:
\( 1458 \div 2 = 729 \)
- Now, 729 is \( 3^6 \) (since \( 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \)).

So, we can express 5832 as:
\[ 5832 = 2^3 \times 3^6 \]

### Step 2: Rewrite the logarithm using the change of base formula
Now, we can rewrite the logarithm using the change of base formula:
\[ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \]

In our case, we can use base 3:
\[ \log_{3\sqrt{2}} 5832 = \frac{\log_{3} 5832}{\log_{3} (3\sqrt{2})} \]

### Step 3: Calculate \( \log_{3} 5832 \)
Using the factorization from Step 1:
\[ \log_{3} 5832 = \log_{3} (2^3 \times 3^6) = 3 \log_{3} 2 + 6 \log_{3} 3 \]
Since \( \log_{3} 3 = 1 \):
\[ \log_{3} 5832 = 3 \log_{3} 2 + 6 \]

### Step 4: Calculate \( \log_{3} (3\sqrt{2}) \)
Now we calculate \( \log_{3} (3\sqrt{2}) \):
\[ \log_{3} (3\sqrt{2}) = \log_{3} 3 + \log_{3} \sqrt{2} \]
\[ = 1 + \frac{1}{2} \log_{3} 2 \]

### Step 5: Substitute back into the logarithm
Now we substitute back into our logarithm:
\[ \log_{3\sqrt{2}} 5832 = \frac{3 \log_{3} 2 + 6}{1 + \frac{1}{2} \log_{3} 2} \]

### Step 6: Simplify the expression
Let \( x = \log_{3} 2 \):
\[ \log_{3\sqrt{2}} 5832 = \frac{3x + 6}{1 + \frac{1}{2}x} \]

To simplify:
Multiply numerator and denominator by 2 to eliminate the fraction in the denominator:
\[ = \frac{2(3x + 6)}{2 + x} = \frac{6x + 12}{2 + x} \]

### Final Answer
Thus, the value of \( \log_{3\sqrt{2}} 5832 \) is:
\[ \frac{6x + 12}{2 + x} \]
where \( x = \log_{3} 2 \).

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