If `5^(5-x)=2^(x-5)` , find the value of `x`

Options

A. 5
B. 0
C. 1
D. can't be determined

Answer

To solve the equation \( 5^{5-x} = 2^{x-5} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 5^{5-x} = 2^{x-5} \]

### Step 2: Take logarithm of both sides
Taking the logarithm of both sides (we can use any logarithm, but here we will use the natural logarithm for simplicity):
\[ \ln(5^{5-x}) = \ln(2^{x-5}) \]

### Step 3: Apply the power rule of logarithms
Using the power rule of logarithms, we can bring down the exponents:
\[ (5-x) \ln(5) = (x-5) \ln(2) \]

### Step 4: Distribute the logarithms
Now, we distribute the logarithms:
\[ 5 \ln(5) - x \ln(5) = x \ln(2) - 5 \ln(2) \]

### Step 5: Rearrange the equation
Rearranging the equation to isolate terms involving \( x \):
\[ 5 \ln(5) + 5 \ln(2) = x \ln(2) + x \ln(5) \]
\[ 5 (\ln(5) + \ln(2)) = x (\ln(2) + \ln(5)) \]

### Step 6: Factor out \( x \)
Factoring out \( x \) from the right side:
\[ 5 (\ln(5) + \ln(2)) = x (\ln(2) + \ln(5)) \]

### Step 7: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{5 (\ln(5) + \ln(2))}{\ln(2) + \ln(5)} \]
Since \( \ln(5) + \ln(2) \) is common in both numerator and denominator, we can simplify:
\[ x = 5 \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{5} \]