Coefficient interpreation of linear model when log2 transforming independent variables I have been advised to log2 transform my independent dietary variables which are intakes of vitamin D and energy intake (kcal).
My research question is whether vitamin D intake affects inflammatory scores, and I adjust for energy intake and other covariates.
I have normalised these inflammatory scores using Box Cox and have this as my dependent variable in a linear regression model like..
Scores ~ age + sex + BMI +  vitamin D (log2 transformed) + energy intake (log2 transformed) + other covariates. 

I have read in these publications :
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6760973/ and
https://pubmed.ncbi.nlm.nih.gov/34804518/
that following log2 transformation, each increase of one unit of these dietary variables corresponds to a doubling in intake. Therefore, the coefficients represent the effect of doubling the amounts of vitamin D and calories.
Please could somebody explain that mathematically, I'm a little unsure of how it represents "doubling"?
 A: First, remember that doubling a number corresponds to an increase of $1$ in its $\log_2$ because $\log_2(2x) - \log_2(x) = \log_2\left(\frac{2x}{x}\right) = \log_2(2) = 1$.
Concerning interpretation of the coefficient, let's start with a simple model:
$$
y=\beta_0 + \beta_1\log_2(x)
$$
Let's calculate the model for a multiplicative increase $k$ of $x$ and compare it to the base model:
$$
y_1=\beta_0 + \beta_1\log_2(x)\\
y_2=\beta_0 + \beta_1\log_2(kx)
$$
Now take the difference of the second and first equation:
$$
y_2-y_1 = (\beta_0 + \beta_1\log_2(kx)) - (\beta_0 + \beta_1\log_2(x)) = \beta_1(\log_2(kx) - \log_2(x)) = \beta_1\log_2\left(\frac{kx}{x}\right) = \beta_1\log_2(k)
$$
Using $k=2$ which represents a doubling of $x$, we have $\beta_1\log_2(2) = \beta_1$. So for a doubling of $x$, $y$ is expected to change by $\beta_1$. This holds only if we use the log to base $2$, of course. Generally, we can summarize the finding as follows:

For an increase of $x$ by a factor of $k$, $y$ changes by
$\beta_1\log_2(k)$.

