What is meant by $\log_2 X \sim N(0,n) \implies \log X \sim N(0, n (\log 2)^2)$? I see the following written in my class notes:

$\log_2 X \sim N(0,n) \implies \log X \sim N(0, n (\log 2)^2)$

But I don't understand what they did here. It looked like they did some kind of conversion of the base on the log, but what is the base on the log on the RHS?

Let $Y = \log_2X \sim N(0, n)$.
By using a formula
$$
\log_2X = \frac{\log X}{\log 2}, \text{ where } \log X \text{ is the natural }\log \text{ of }X,
$$
one obtains that
$$
\log X = \log{2}\cdot \log_2{X} = \log{2}\cdot Y.
$$
Then, we can use a property of the Normal distribution
$$
\text{If }Y \sim N(\mu, \sigma^2), \text{ then }aY\sim N(a\mu, a^2\sigma^2). 
$$
In our case, we have
$$
Y = \log_2X \sim N(0, n), a = \log{2}\Rightarrow aY = \log{X} \sim N(0, n(\log{2})^2). 
$$
