Find the logarithm of 144 to the base `2sqrt(3)`.

Options

A. 8
B. 4
C. `2sqrt(3)`
D. none of these

Answer

To find the logarithm of 144 to the base \(2\sqrt{3}\), we can follow these steps:

### Step-by-Step Solution:

1. **Set Up the Logarithmic Equation**:
We want to find \( \log_{2\sqrt{3}}{144} \). Let's denote this value as \( x \).
\[ x = \log_{2\sqrt{3}}{144} \]

2. **Convert to Exponential Form**:
Using the property of logarithms, we can convert this logarithmic equation into its equivalent exponential form:
\[ 144 = (2\sqrt{3})^x \]

3. **Express 144 in Terms of Its Prime Factors**:
We know that \( 144 = 12^2 \) and \( 12 = 2^2 \cdot 3 \). Therefore:
\[ 144 = (2^2 \cdot 3)^2 = 2^4 \cdot 3^2 \]

4. **Express the Base \(2\sqrt{3}\)**:
The base can be expressed as:
\[ 2\sqrt{3} = 2 \cdot 3^{1/2} \]
Thus,
\[ (2\sqrt{3})^x = (2 \cdot 3^{1/2})^x = 2^x \cdot 3^{x/2} \]

5. **Set Up the Equation**:
Now we can equate the two expressions:
\[ 2^4 \cdot 3^2 = 2^x \cdot 3^{x/2} \]

6. **Compare the Exponents**:
Since the bases are the same, we can set the exponents equal to each other:
- For base \(2\):
\[ x = 4 \]
- For base \(3\):
\[ \frac{x}{2} = 2 \implies x = 4 \]

7. **Conclusion**:
Both equations give us the same result. Therefore, the value of \( x \) is:
\[ \log_{2\sqrt{3}}{144} = 4 \]

### Final Answer:
\[ \log_{2\sqrt{3}}{144} = 4 \]