Question

Let f : R⁺ → R, where R⁺ is the set of all positive real numbers, such that f(x) = log_e x. Determine

(c) whether f(xy) = f(x) : f(y) holds

 

Answer 1

Given:
f : R⁺ → R
and f (x) = log_ex .............(i)

(c) f (xy) = log_e(xy)      {From(i)}
         = log_ex + log_ey         [Since log_emn = log_e m + log_en]  
         = f (x) + f (y)
Thus, f (xy) = f (x) + f (y)
Hence, it is clear that f (xy) = f (x) + f (y) holds.