STATEMENT-1 : `log_(4) 16` is a rational number . And
STATEMENT-2 : ` log_(2) 3 ` is an irrational number

Options

A. Statemant-1 is True , Statement-2 is True, Statement -2 is a correct explanation for Statement-1
B. Statemant-1 is True , Statement-2 is True, Statement -2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Stetement-2 is False.
D. Statement-1 is False, Statement-2 is True

Answer

To solve the question, we need to analyze both statements provided:

**Statement 1:** `log_(4) 16` is a rational number.

**Statement 2:** `log_(2) 3` is an irrational number.

Let's evaluate each statement step by step.

### Step 1: Evaluate Statement 1

We need to determine if `log_(4) 16` is a rational number.

1. **Convert the logarithm to a simpler form:**
\[ \log_{4} 16 = \frac{\log_{2} 16}{\log_{2} 4} \]
Using the change of base formula.

2. **Calculate `log_{2} 16`:**
\[ 16 = 2^4 \implies \log_{2} 16 = 4 \]

3. **Calculate `log_{2} 4`:**
\[ 4 = 2^2 \implies \log_{2} 4 = 2 \]

4. **Substitute back into the logarithm:**
\[ \log_{4} 16 = \frac{4}{2} = 2 \]

5. **Determine if the result is rational:**
Since \(2\) can be expressed as \(\frac{2}{1}\), it is a rational number.

### Conclusion for Statement 1:
`log_(4) 16` is indeed a rational number.

### Step 2: Evaluate Statement 2

Now we need to determine if `log_(2) 3` is an irrational number.

1. **Assume `log_(2) 3` is rational:**
If `log_(2) 3` is rational, then it can be expressed as \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\).

2. **Convert the logarithm to exponential form:**
\[ 3 = 2^{\frac{p}{q}} \implies 3^q = 2^p \]

3. **Analyze the equation:**
The left side \(3^q\) is an odd number (since 3 is odd), and the right side \(2^p\) is an even number (since 2 is even).

4. **Conclusion:**
An odd number cannot equal an even number. Thus, our assumption that `log_(2) 3` is rational must be incorrect.

### Conclusion for Statement 2:
`log_(2) 3` is indeed an irrational number.

### Final Conclusion:
Both statements are true:
- Statement 1 is true: `log_(4) 16` is a rational number.
- Statement 2 is true: `log_(2) 3` is an irrational number.

However, Statement 2 does not provide a correct explanation for Statement 1.

### Answer:
The correct option is:
**Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.**

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