# Proving an inequality involving logarithm

I have the following inequality:

$$(1/2)log_2n >= 1$$

which I am trying to prove. More specifically, I am trying to find an integer N > 0 for which the inequality is true for all n >= N.

I know from substituting values, the inequality holds for all n >= 4.

However, I would like to prove the inequality in a better way. After all, you can't guarantee that the inequality holds for arbitrarily large values of n just from substituting values. What would be some techniques that I can use to prove that the inequality holds for all n >= N? Any insights are appreciated.

It is $$\frac{\ln(n)}{\ln(2)}\geq 2$$ so $$\ln(n) \geq 2\ln(2)$$ $$n\geq e^{2\ln(2)}=e^{\ln(4)}=4$$
• It is $$\ln(x^r)=r\ln(x)$$ for $$x>0$$ – Dr. Sonnhard Graubner Mar 10 at 7:51