TITLE: If $f(x)$ satisfies $2f (x) = f(xy) + f(x/y)$, find $f(x)$
QUESTION [9 upvotes]: If $f(x)$ is a continuous and differentiable function which satisfies the function equation If $$2f (x) = f(xy) + f\left(\frac xy\right)\quad \forall x,y \in \mathbb{R}^{+}$$ and $f'(1)=1$ then find $f(x)$
I can see that $f(x)=\ln(x)$ is one function which satisfies all the properties but how can it be proved? I tried using first principle of differentiation but wasn't able to obtain the function. Could someone help me with this?

REPLY [4 votes]: Problem:   Find all continuous  functions $f:\Bbb{R}^+\to\Bbb{R}$ which satisfy
  $$ \forall_{x\ y>0}\quad 2\cdot f(x)
     \,\ =\,\ f(x\cdot y)\ +\ f\left(\dfrac{x}y\right).$$

Solution:
Let $\ s\ :=\ x\cdot y\ $ and $\ t\ :=\ \frac xy;\ \ $
i.e. $\ \ x\ =\ \sqrt{s\cdot t}\ $ and $\ y\ =\ \sqrt{\frac st}.\ $
Then the above equation is equivalent to:
$$ f\left(\sqrt{s\cdot t}\right)\,\ =\,\ \frac {f(s)+f(t)}2 $$
Define function
$$ g:=f\circ exp: \mathbb R\rightarrow\mathbb R $$
Then:
$$ \forall_{\sigma\ \tau\in\mathbb R}\quad g\left(\frac{\sigma+\tau}2\right)\ =\ \frac{g(\sigma)+g(\tau)}2 $$
It's known that such $\ g\ $ is a linear function (meaning, a polynomial of degree $1$), i.e. there exist $\ A\ B\in\mathbb R\ $ such that:
$$ g(\xi)\ =\ A\cdot\xi+B $$
hence
$$ f(x)\ =\ (f\circ\exp\circ\log)(x)\ =\ (g\circ\log)(x)\ =\ A\cdot\log(x)+B $$

Solution:   The set of required functions $\ f\ $ is
  $$ \{ A\!\cdot\!\log\, +\, B\ :\,\ A\,\ B\in\mathbb R\} $$