Evaluate :
` (log 27)/( log sqrt3)`

Answer

To evaluate \( \frac{\log 27}{\log \sqrt{3}} \), we can follow these steps:

### Step 1: Rewrite the logarithms
We know that \( 27 \) can be expressed as \( 3^3 \) and \( \sqrt{3} \) can be expressed as \( 3^{1/2} \). Therefore, we can rewrite the logarithms as follows:
\[ \log 27 = \log(3^3) \quad \text{and} \quad \log \sqrt{3} = \log(3^{1/2}) \]

### Step 2: Apply the power rule of logarithms
Using the property of logarithms that states \( \log(a^b) = b \cdot \log a \), we can simplify both logarithms:
\[ \log(3^3) = 3 \log 3 \quad \text{and} \quad \log(3^{1/2}) = \frac{1}{2} \log 3 \]

### Step 3: Substitute back into the expression
Now we can substitute these simplified forms back into our original expression:
\[ \frac{\log 27}{\log \sqrt{3}} = \frac{3 \log 3}{\frac{1}{2} \log 3} \]

### Step 4: Simplify the fraction
The \( \log 3 \) in the numerator and denominator cancels out:
\[ \frac{3 \log 3}{\frac{1}{2} \log 3} = \frac{3}{\frac{1}{2}} = 3 \times 2 = 6 \]

### Final Answer
Thus, the value of \( \frac{\log 27}{\log \sqrt{3}} \) is \( 6 \).

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