Question

已知f(x)的定义域为(0,正无穷)且f(xy)=f(x)+f(y)求证f(x^m)=mf(x)

Answer 1

f(x^m) = f(x^(m-1)*x) (这步是因为x^m = x^m-1 * x)= f(x^(m-1)) + f(x) (这步是利用f(xy)=f(x)+f(y))

再对f(x^(m-1))做同样的分解,f(x^m) = f(x^(m-1)) + f(x) = f(x^(m-2)) + 2f(x),一直下去

明显的f(x^m)=mf(x)