The value of `(1)/(log_(3)pi) + (1)/(log_(4)pi)` is

Options

A. greater than 2
B. less than 2
C. equal to `(1)/(2)`
D. equal to 0

Answer

To solve the expression \(\frac{1}{\log_3 \pi} + \frac{1}{\log_4 \pi}\), we will use the change of base formula for logarithms. The change of base formula states that:

\[ \log_a b = \frac{\log_c b}{\log_c a} \]

for any positive \(c\). We will use natural logarithms (base \(e\)) for our calculations.

### Step 1: Apply the change of base formula

Using the change of base formula, we can rewrite the logarithms:

\[ \log_3 \pi = \frac{\log \pi}{\log 3} \]
\[ \log_4 \pi = \frac{\log \pi}{\log 4} \]

### Step 2: Substitute into the expression

Now we can substitute these into our original expression:

\[ \frac{1}{\log_3 \pi} = \frac{\log 3}{\log \pi} \]
\[ \frac{1}{\log_4 \pi} = \frac{\log 4}{\log \pi} \]

Thus, the expression becomes:

\[ \frac{\log 3}{\log \pi} + \frac{\log 4}{\log \pi} \]

### Step 3: Combine the fractions

Since both terms have a common denominator, we can combine them:

\[ \frac{\log 3 + \log 4}{\log \pi} \]

### Step 4: Use the property of logarithms

We can use the property of logarithms that states \(\log a + \log b = \log(ab)\):

\[ \log 3 + \log 4 = \log(3 \cdot 4) = \log 12 \]

So, we can rewrite our expression as:

\[ \frac{\log 12}{\log \pi} \]

### Step 5: Rewrite using the change of base formula again

We can again apply the change of base formula to express this in terms of base \(\pi\):

\[ \frac{\log 12}{\log \pi} = \log_\pi 12 \]

### Step 6: Analyze the value of \(\log_\pi 12\)

To evaluate \(\log_\pi 12\), we need to compare it to 2. We know that:

- \(\pi^2 \approx 9.87\), which is less than 12.
- Therefore, \(\log_\pi 12\) will be greater than 2 because \(12 > \pi^2\).

Thus, we conclude that:

\[ \log_\pi 12 > 2 \]

### Final Answer

The value of \(\frac{1}{\log_3 \pi} + \frac{1}{\log_4 \pi}\) is greater than 2.

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