How Can This Be Proven By Induction?

Question

I cannot find the "path" to follow to demonstrate that this is verified as n=k and when n=k+1 .

$\ln(n) < n$

Thank you in advance, your help is really appreciated.

Answer 1

Hint. Write $n=e^{\ln(n)}=e^k$ and prove $k<e^k$ by induction on $k$.

Answer 2

Base case: $0=\ln(1)< 1$.

Induction: for $k \ge 1$, $$\ln(k+1)-\ln(k) = \ln(1+1/k) \le \ln(e) \le 1$$ so $$(k+1)-\ln(k+1) = (k - \ln(k)) + (1 - \ln(k+1)+\ln(k)) \ge k-\ln(k) > 0.$$