[Math] Prove $\log(n) = O(n)$ using induction

Question

I am using the lecture notes here on page 19 (Algorithm Notes 1) example 1 is the inductive proof of

$\log(n) = O(n)$.

I understand the base case but I don't understand the rest of the example.

Example: $\log(n) = O(n)$. Claim: For all $n \geq 1$, $\log(n) \leq n$. The proof is by induction on $n$. The claim is trivially true for $n=1$, since $0<1$.

Now suppose $n\geq 1$ and $\log(n) \leq n$. Then $$ \log(n+1) \\ \leq \log(2n) \\ = \log(n) + 1 \\ \leq n+1 \quad \text{by the inductive hypothesis}.$$

I need help understanding how

$\log(n + 1) \leq \log(2n)$.

I don't get where the $\log(2n)$ came from.

Answer 1

$n + 1 \le 2n \implies \log(n+1) \le \log(2n)$