If `3f(x)-f((1)/(x))= log_(e) x^(4)` for `x gt 0` ,then
`f(e^(x))=`

Options

A. x
B. `log_(e)x `
C. `e^(x)`
D. none of these

Explanation

To solve the equation \(3f(x) - f\left(\frac{1}{x}\right) = \log_e x^4\) for \(x > 0\) and find \(f(e^x)\), we will follow these steps:

### Step 1: Rewrite the given equation
We start with the equation:
\[ 3f(x) - f\left(\frac{1}{x}\right) = \log_e x^4 \]
Using the property of logarithms, we can rewrite \(\log_e x^4\) as:
\[ \log_e x^4 = 4 \log_e x \]
Thus, we can rewrite our equation as:
\[ 3f(x) - f\left(\frac{1}{x}\right) = 4 \log_e x \tag{1} \]

### Step 2: Substitute \(x\) with \(\frac{1}{x}\)
Now we will substitute \(x\) with \(\frac{1}{x}\) in equation (1):
\[ 3f\left(\frac{1}{x}\right) - f(x) = 4 \log_e \left(\frac{1}{x}\right) \]
Using the property of logarithms again, we know that \(\log_e \left(\frac{1}{x}\right) = -\log_e x\). Therefore, we can rewrite the equation as:
\[ 3f\left(\frac{1}{x}\right) - f(x) = -4 \log_e x \tag{2} \]

### Step 3: Multiply equation (1) by 3
Now, we will multiply equation (1) by 3:
\[ 9f(x) - 3f\left(\frac{1}{x}\right) = 12 \log_e x \tag{3} \]

### Step 4: Add equations (2) and (3)
Next, we will add equations (2) and (3):
\[ (9f(x) - 3f\left(\frac{1}{x}\right)) + (3f\left(\frac{1}{x}\right) - f(x)) = 12 \log_e x - 4 \log_e x \]
This simplifies to:
\[ 8f(x) = 8 \log_e x \]
Dividing both sides by 8 gives us:
\[ f(x) = \log_e x \tag{4} \]

### Step 5: Find \(f(e^x)\)
Now, we need to find \(f(e^x)\). We substitute \(x\) with \(e^x\) in equation (4):
\[ f(e^x) = \log_e (e^x) \]
Using the property of logarithms, we know that:
\[ \log_e (e^x) = x \]
Thus, we conclude:
\[ f(e^x) = x \]

### Final Answer
The final answer is:
\[ f(e^x) = x \]