About functional equation $f(st)=f(s)+f(t)(s,t\in \mathbb R\backslash \{0\})$


Ramiz Vugdalić, Fatih Destović




We analyse continuously differentiable functions $f:\mathbb R\backslash\{0\}\to\mathbb R$, that are the solutions of functional equation $f(st)=f(s)+f(t)$. We prove that $f\equiv 0$, and logarithmic functions $f(t)=\log_a|t|$, $(0<a\neq1)$ are the only solutions of the equation above.