Answer :

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]