I have a question that says the following, (it is meant to be solved using Lagrange Mean Value Theorem):
If $f(x)=\log_ex$, $g(x) = x^2$ and $c \in (4,5)$, then $c\log_e\bigg(\frac{4^{25}}{5^{16}}\bigg)$ is equal to which of the following value? (The options are given in the book). The solution says that the answer is $2(c^2\log_e4-8)$
Now, I looked at the solution of this question and in the first line itself, the book has assumed to function on which LMVT is applied to be
$$\phi(x)=x^2\log_e4-16\log_ex$$
I have no clue how this is brought up. How did they come up with this specific function? However, I noticed that from the given question, I could simplify the required quantity as $$c(5^2\log_e4-4^2\log_e5)$$
which looks an awful lot like the function that they came up with. So, my question is, why did they take the terms which contain the "$5$'s" to be the variable $x$, and why not, say the "$4$'s". Could I have taken the function as $h(x)= 25\log_ex-x^2\log_e5$?
Moreover, how did they even come up with this function? What is the thought process that goes into it and how do I 'get' that in an examination mindset? Some more examples or references on how this works would be nice (I'm a high schooler). Thanks!