Let f: R+→ R, where R+ is the set of all positive real numbers, be such that f(x) = loge x. Determine whether f (xy) = f (x) + f (y) holds.

Answer:

Given,

f (x) = log_e x ⇒ f (y) = log_e y

Consider,

f (xy)

F (xy) = log_e (xy)

f (xy) = log_e (x × y) [since, log_b (a×c) = log_b a + log_b c]

f (xy) = log_e x + log_e y

∴ f (xy) = f (x) + f (y)