If `((a)/(b))^(x-10) = ((b)/(a))^(x-16)`, then `x` is equal to

Answer

To solve the equation \(\left(\frac{a}{b}\right)^{x-10} = \left(\frac{b}{a}\right)^{x-16}\), we can follow these steps:

### Step 1: Rewrite the Right Side
We can rewrite the right side of the equation \(\left(\frac{b}{a}\right)^{x-16}\) in terms of \(\left(\frac{a}{b}\right)\):
\[ \left(\frac{b}{a}\right)^{x-16} = \left(\frac{a}{b}\right)^{-(x-16)} \]
This is because \(\frac{b}{a} = \frac{1}{\frac{a}{b}}\).

### Step 2: Set the Exponents Equal
Now we have:
\[ \left(\frac{a}{b}\right)^{x-10} = \left(\frac{a}{b}\right)^{-(x-16)} \]
Since the bases are the same, we can equate the exponents:
\[ x - 10 = -(x - 16) \]

### Step 3: Simplify the Equation
Now, simplify the equation:
\[ x - 10 = -x + 16 \]
Add \(x\) to both sides:
\[ x + x - 10 = 16 \]
This simplifies to:
\[ 2x - 10 = 16 \]

### Step 4: Solve for \(x\)
Now, add 10 to both sides:
\[ 2x = 26 \]
Finally, divide by 2:
\[ x = 13 \]

### Conclusion
Thus, the value of \(x\) is:
\[ \boxed{13} \]

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