[Math] Show that for every real number x, there exists a natural number n such that $x < 2^n$

Question

Let x be a positive real number. I want to prove that $\forall$ x, $\exists$ n $\in$ N such that x < $2^n$ .

To me it seems that as x increases, I can just pick larger and larger values for n to satisfy this property. Since n goes to infinity, I should be able to do this process forever. Any idea how to prove this?

Answer 1

HINT: Use logarithm and the Archimedean Property of the real numbers.