Answer: The answer is f(x) = -7x + 7.
Step-by-step explanation: We are give a relation as follows :
[tex]5^{-4x+7}\div 125^x=5^{f(x)}.[/tex]
From here, we need to find the expression for f(x).
Here, we will be using the following properties of exponents :
[tex](i)~\dfrac{a^x}{a^y}=a^{x-y}.\\\\(ii)~a^x=a^y~\Rightarrow x=y.[/tex]
We have
[tex]5^{-4x+7}\div 125^x=5^{f(x)}\\\\\Rightarrow5^{-4x+7}\div 5^{3x}=5^{f(x)}\\\\\\\Rightarrow \dfrac{5^{-4x+7}}{5^{3x}}=5^{f(x)}\\\\\\\Rightarrow 5^{-4x+7-3x}=5^{f(x)}}\\\\\Rightarrow 5^{-7x+7}=5^{f(x)}\\\\\Rightarrow -7x+7=f(x).[/tex]
Thus, the required expression is f(x) = -7x + 7.
