proof writing - Prove the inequality for all natural numbers n using induction

$\log_2 n

I know how to prove the base case Base Case $\log_2 1<1$

likewise assuming the inequality for n=k; $\log_2 k

Then to prove by induction I show $\log_2 k<(k+1)$?

I know it's true since the domain is all real numbers i just cant figure out the next step to prove it.

Answer

$\log_2 (n) = \frac{\ln (n)}{\ln(2)}$, where $ln$ is the natural log i.e. with base $e$

You need to show, $\log_2(n) < n\implies \frac{\ln (n)}{\ln(2)} < n \implies \ln(n)

Since $\log$ is an increasing function you can rephrase your question to $n < 2^n$ as suggested by @jwsiegel

That is easy to show by induction

n=1: $1<2^1$. True.

Induction Hypothesis: $n-1< 2^{n-1}$

for $n \geq 2$, $n \leq 2(n-1) < 2*2^{n-1} = 2^n$