Using mathematical induction to show $\log(n) \leq \log(2n)$

Question

Using the principle of mathematical induction, prove that $\log(1) \leq \log(2) \leq \log(n) \leq \log(2n)$ for all $n$.

Answer 1

$\text{log} (2) = \text{log}(2 \times 1) = \text{log} (1) + \text{log}(2) \geq \text{log}(1)$

$\text{log} (2n) = \text{log}(2 \times n) = \text{log}(2) + \text{log}(n) \geq \text{log}(n)$

(Given the base is $> 1$)

In general:
$\text{log}(2) \text{ is not} \leq \text{log}(n)$