Basic Analysis I — Real Analysis · Problem 5.4.10

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · uniqueness-exponential-logarithm

Problem

Show that

E(x) = e^x

is the unique continuous function such that

E(x+y) = E(x)E(y)

and

E(1) = e

. Similarly, prove that

L(x) = \ln(x)

is the unique continuous function defined on positive

x

such that

L(xy) = L(x) + L(y)

and

L(e) = 1