Question

设f(x)在(0,∞)内有定义,且f '(1)=a,另在(0,∞)内,对任意的x,y,f(xy)=f(x)+f(y),求

Answer 1

对f(xy)=f(x)+f(y)两边同时对x求导,

得到yf'(xy)=f'(x),

令x=1,则yf'(y)=f'(1)=a,

再用x换y,得到f'(x)=a/x,两边积分,

得到f(x)=alnx.