If `3^(((log_3 7))^x)=7^(((log_7 3))^x)`, then the value of x will be

Options

A. `(1)/(2)`
B. `(1)/(4)`
C. `(1)/(3)`
D. 1

Explanation

To solve the equation \( 3^{(\log_3 7)^x} = 7^{(\log_7 3)^x} \), we can follow these steps:

### Step 1: Take logarithm on both sides
We start by taking the logarithm of both sides of the equation. We can use any logarithm base, but for simplicity, we will use the natural logarithm (ln):

\[ \log(3^{(\log_3 7)^x}) = \log(7^{(\log_7 3)^x}) \]

### Step 2: Apply the power rule of logarithms
Using the power rule of logarithms, which states that \(\log(a^b) = b \cdot \log(a)\), we can rewrite both sides:

\[ (\log_3 7)^x \cdot \log(3) = (\log_7 3)^x \cdot \log(7) \]

### Step 3: Change of base formula
Next, we apply the change of base formula for logarithms, which states that \(\log_a b = \frac{\log b}{\log a}\):

\[ \left(\frac{\log 7}{\log 3}\right)^x \cdot \log(3) = \left(\frac{\log 3}{\log 7}\right)^x \cdot \log(7) \]

### Step 4: Rearranging the equation
Now, we can rearrange the equation to isolate the terms involving \(x\):

\[ \frac{(\log 7)^x}{(\log 3)^x} \cdot \log(3) = \frac{(\log 3)^x}{(\log 7)^x} \cdot \log(7) \]

### Step 5: Cross-multiplying
Cross-multiplying gives us:

\[ (\log 7)^{x+1} = (\log 3)^{x+1} \]

### Step 6: Setting the bases equal
Since the bases are equal, we can equate the exponents:

\[ x + 1 = x + 1 \]

### Step 7: Solving for \(x\)
Now, we can simplify this to find \(x\):

\[ 2x = 1 \]

Dividing both sides by 2 gives:

\[ x = \frac{1}{2} \]

### Final Answer
Thus, the value of \(x\) is:

\[ \boxed{\frac{1}{2}} \]