If `(1)/("log"_(3) pi) + (1)/("log"_(4) pi) gt x`, then the greatest integral value of is

Options

A. 2
B. 3
C. `pi`
D. none of these

Answer

To solve the inequality \( \frac{1}{\log_3 \pi} + \frac{1}{\log_4 \pi} > x \), we can follow these steps:

### Step 1: Rewrite the logarithms
Using the change of base formula, we can rewrite the logarithms:
\[ \log_3 \pi = \frac{1}{\log_\pi 3} \quad \text{and} \quad \log_4 \pi = \frac{1}{\log_\pi 4} \]
Thus, we have:
\[ \frac{1}{\log_3 \pi} = \log_\pi 3 \quad \text{and} \quad \frac{1}{\log_4 \pi} = \log_\pi 4 \]

### Step 2: Combine the logarithms
Now we can combine the two logarithms:
\[ \log_\pi 3 + \log_\pi 4 = \log_\pi (3 \cdot 4) = \log_\pi 12 \]
So, the inequality becomes:
\[ \log_\pi 12 > x \]

### Step 3: Exponentiate to eliminate the logarithm
Using the property of logarithms, we can exponentiate both sides:
\[ 12 > \pi^x \]
This can be rearranged to:
\[ \pi^x < 12 \]

### Step 4: Take logarithm of both sides
Taking the logarithm (base \( e \) or natural logarithm) of both sides gives us:
\[ \log(\pi^x) < \log(12) \]
This simplifies to:
\[ x \cdot \log(\pi) < \log(12) \]

### Step 5: Solve for \( x \)
Dividing both sides by \( \log(\pi) \) (noting that \( \log(\pi) > 0 \)):
\[ x < \frac{\log(12)}{\log(\pi)} \]

### Step 6: Calculate the values
Now we need to calculate \( \log(12) \) and \( \log(\pi) \):
- \( \log(12) \approx 2.4849 \)
- \( \log(\pi) \approx 1.1447 \)

Now substituting these values:
\[ x < \frac{2.4849}{1.1447} \approx 2.171 \]

### Step 7: Find the greatest integral value of \( x \)
The greatest integral value of \( x \) that satisfies this inequality is:
\[ \lfloor 2.171 \rfloor = 2 \]

Thus, the greatest integral value of \( x \) is \( 2 \).