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: Use the Change of Base Formula
We start by applying the change of base formula for logarithms, which states that:
\[ \frac{1}{\log_b a} = \log_a b \]
Using this, we can rewrite the terms:
\[ \frac{1}{\log_3 \pi} = \log_\pi 3 \quad \text{and} \quad \frac{1}{\log_4 \pi} = \log_\pi 4 \]
Thus, the inequality becomes:
\[ \log_\pi 3 + \log_\pi 4 > x \]

### Step 2: Combine the Logarithms
Using the property of logarithms that states \( \log_a b + \log_a c = \log_a (bc) \), we can combine the logarithms:
\[ \log_\pi 3 + \log_\pi 4 = \log_\pi (3 \cdot 4) = \log_\pi 12 \]
So, we now have:
\[ \log_\pi 12 > x \]

### Step 3: Rewrite the Inequality
This can be rewritten as:
\[ x < \log_\pi 12 \]

### Step 4: Change the Base Again
Using the change of base formula again, we can express \( \log_\pi 12 \) in terms of natural logarithms:
\[ \log_\pi 12 = \frac{\log 12}{\log \pi} \]

### Step 5: Calculate the Value
Now we need to find the numerical value of \( \frac{\log 12}{\log \pi} \). Using a calculator:
- \( \log 12 \approx 1.07918 \)
- \( \log \pi \approx 1.14473 \)

Thus,
\[ \log_\pi 12 \approx \frac{1.07918}{1.14473} \approx 0.942 \]

### Step 6: Find the Greatest Integral Value
Since \( x < \log_\pi 12 \approx 0.942 \), the greatest integral value of \( x \) that satisfies this inequality is:
\[ \lfloor 0.942 \rfloor = 0 \]

### Conclusion
Therefore, the greatest integral value of \( x \) is \( 0 \).