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 \( x > 0 \), we will follow these steps:

### Step 1: Simplify the Right Side
First, we simplify the right side of the equation:
\[ \log_e(x^4) = 4 \log_e(x) \]
So the equation becomes:
\[ 3f(x) - f\left(\frac{1}{x}\right) = 4 \log_e(x) \]

### Step 2: Substitute \( x \) with \( \frac{1}{x} \)
Next, we substitute \( x \) with \( \frac{1}{x} \) in the original equation:
\[ 3f\left(\frac{1}{x}\right) - f(x) = \log_e\left(\left(\frac{1}{x}\right)^4\right) \]
This simplifies to:
\[ 3f\left(\frac{1}{x}\right) - f(x) = \log_e\left(\frac{1}{x^4}\right) = -4 \log_e(x) \]

### Step 3: Set Up the System of Equations
Now we have two equations:
1. \( 3f(x) - f\left(\frac{1}{x}\right) = 4 \log_e(x) \) (Equation 1)
2. \( 3f\left(\frac{1}{x}\right) - f(x) = -4 \log_e(x) \) (Equation 2)

### Step 4: Solve the System of Equations
From Equation 1, we can express \( f\left(\frac{1}{x}\right) \):
\[ f\left(\frac{1}{x}\right) = 3f(x) - 4 \log_e(x) \]
Now substitute this expression into Equation 2:
\[ 3(3f(x) - 4 \log_e(x)) - f(x) = -4 \log_e(x) \]
This simplifies to:
\[ 9f(x) - 12 \log_e(x) - f(x) = -4 \log_e(x) \]
Combining like terms gives:
\[ 8f(x) - 12 \log_e(x) = -4 \log_e(x) \]
Adding \( 12 \log_e(x) \) to both sides:
\[ 8f(x) = 8 \log_e(x) \]
Dividing both sides by 8:
\[ f(x) = \log_e(x) \]

### Step 5: Find \( f(e^x) \)
Now we need to find \( f(e^x) \):
\[ f(e^x) = \log_e(e^x) = x \]

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

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