The value of `log_(2sqrt(3))(1728)` is

Options

A. A)3
B. B)5
C. C)6
D. D)9

Answer

To find the value of \( \log_{2\sqrt{3}}(1728) \), we can follow these steps:

### Step 1: Rewrite the logarithm
We start by expressing \( 1728 \) in terms of its prime factors.
\[ 1728 = 12^3 \]
This is because \( 12 = 2^2 \cdot 3 \) and \( 12^3 = (2^2 \cdot 3)^3 = 2^6 \cdot 3^3 = 1728 \).

### Step 2: Change the base
Now we can rewrite the logarithm:
\[ \log_{2\sqrt{3}}(1728) = \log_{2\sqrt{3}}(12^3) \]

### Step 3: Use the power rule of logarithms
Using the power rule of logarithms, we can bring down the exponent:
\[ \log_{2\sqrt{3}}(12^3) = 3 \cdot \log_{2\sqrt{3}}(12) \]

### Step 4: Simplify the base
Next, we can express \( 2\sqrt{3} \) as \( \sqrt{12} \):
\[ 2\sqrt{3} = \sqrt{12} \]

### Step 5: Apply the change of base formula
Now we can rewrite the logarithm:
\[ 3 \cdot \log_{\sqrt{12}}(12) \]

### Step 6: Evaluate the logarithm
Using the property of logarithms, we know that:
\[ \log_{a}(a) = 1 \]
Thus:
\[ \log_{\sqrt{12}}(12) = 1 \]

### Step 7: Final calculation
Now substituting back:
\[ 3 \cdot 1 = 3 \]

### Conclusion
Therefore, the value of \( \log_{2\sqrt{3}}(1728) \) is:
\[ \boxed{6} \]

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