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?

Recall from algebra, $a^{\log_a b} = b \to \log_a b \ln a= \ln b \to \log_a b = \frac{\ln b}{\ln a}.$ So, we can rewrite our variable $log_2 X$ as $\frac{1}{\ln 2} \ln X.$
We know that if we have a variable $A \sim N(\mu, \sigma^2)$ then $cA \sim N(c\mu, c^2\sigma^2).$
So, if $log_2 X \sim N(0, n),$ then $\ln2 * \log_2 X = \ln X \sim N(0, n(\ln 2)^2).$
