To determine the value of [tex]\( x \)[/tex] that satisfies the equation
[tex]\[
\left(\frac{m}{n}\right)^{x-1} = \left(\frac{n}{m}\right)^{x-5},
\][/tex]
we will proceed step-by-step.
1. Understanding the Equation:
The equation given is:
[tex]\[
\left(\frac{m}{n}\right)^{x-1} = \left(\frac{n}{m}\right)^{x-5}.
\][/tex]
2. Expressing the Right-Hand Side:
We know that
[tex]\[
\left(\frac{n}{m}\right)^{x-5} = \left(\frac{1}{\frac{m}{n}}\right)^{x-5} = \left(\frac{m}{n}\right)^{-(x-5)}.
\][/tex]
Therefore, the equation becomes:
[tex]\[
\left(\frac{m}{n}\right)^{x-1} = \left(\frac{m}{n}\right)^{-(x-5)}.
\][/tex]
3. Equating the Exponents:
Since the bases are the same [tex]\(\left(\frac{m}{n}\right)\)[/tex], we can equate the exponents:
[tex]\[
x-1 = -(x-5).
\][/tex]
4. Solving for [tex]\( x \)[/tex]:
Now, solve the equation [tex]\( x-1 = -(x-5) \)[/tex]:
[tex]\[
x - 1 = -x + 5.
\][/tex]
Combine like terms:
[tex]\[
x + x = 5 + 1,
\][/tex]
[tex]\[
2x = 6.
\][/tex]
Divide both sides by 2:
[tex]\[
x = 3.
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is
[tex]\[
\boxed{3}.
\][/tex]
