how we proof f(x*y)=f(x)+f(y) this implies f(x)=k*ln(x)

by partial differentiation

differentiate wrt to x

f''(xy)*y=f''(x)

differentiate wrt to y

f''(xy)*x=f''(y)

divide we get    f''(x)x =f''(y)y =k

integrate the first one then weget f(x)=k*ln(x)