Find the logarithm of 144 to the base `2sqrt3`:

Options

A. a) 8
B. b) 4
C. c) `2sqrt3`
D. d) none of these

Answer

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

### Step 1: Set up the logarithmic equation
Let \(x = \log_{2\sqrt{3}}{144}\). By the definition of logarithms, this means:
\[ 144 = (2\sqrt{3})^x \]

### Step 2: Rewrite the base
We can rewrite \(2\sqrt{3}\) as:
\[ 2\sqrt{3} = 2 \cdot 3^{1/2} \]
Thus, we can express the equation as:
\[ 144 = (2 \cdot 3^{1/2})^x \]

### Step 3: Expand the right side
Using the properties of exponents, we can expand the right side:
\[ (2 \cdot 3^{1/2})^x = 2^x \cdot (3^{1/2})^x = 2^x \cdot 3^{x/2} \]
So now we have:
\[ 144 = 2^x \cdot 3^{x/2} \]

### Step 4: Factor 144
Next, we need to express 144 in terms of its prime factors:
\[ 144 = 12^2 = (2^2 \cdot 3)^2 = 2^4 \cdot 3^2 \]
Thus, we can rewrite our equation as:
\[ 2^4 \cdot 3^2 = 2^x \cdot 3^{x/2} \]

### Step 5: Set up equations for the exponents
Now, we can equate the exponents of the same bases:
1. For base 2:
\[ 4 = x \]
2. For base 3:
\[ 2 = \frac{x}{2} \]

### Step 6: Solve for \(x\)
From the first equation, we already have:
\[ x = 4 \]
From the second equation, multiplying both sides by 2 gives:
\[ x = 4 \]

### Conclusion
Both equations confirm that:
\[ x = 4 \]
Thus, the logarithm of 144 to the base \(2\sqrt{3}\) is:
\[ \log_{2\sqrt{3}}{144} = 4 \]