Interpreting a log discrete input variable in a multiple linear regression model I have a multiple linear regression model that has several input variables including a discrete one which has been logged.
$$
\hat{y} = \beta_1*X_1 + ... +\beta_{discrete}*log(X_{discrete}) + ...
$$
I understand that if all other variables are held constant, a 1% increase in $X_{discrete}$ results in a $\beta_{discrete}$ increase in $y$. But a 1% increase in a discrete variable doesn't really make sense.
Is there a way to interpret the impact of a one unit increase in $X_{discrete}$ on $y$?
Thanks
 A: Your same question applies in a simple linear regression model, i.e. 
$$ 
E[Y|X] = \alpha + \beta * \log(X)
$$
The answer to your question is no, there is not one single interpretation. This is because the impact of a one-unit increase in X on the expected value of Y will depend upon the initial value of X. For example, if $X=1$, then a one-unit increase in X will change the expected value from $E[Y|X=1] = \alpha$ to $E[Y|X=2]=\alpha + \beta\log(2)$, for a net change of $\beta\log(2)$. But if $X=10$, then a one-unit increase in X will change the expected value from $E[Y|X=10] = \alpha + \beta\log(10)$ to $E[Y|X=11]=\alpha + \beta\log(11)$, for a net change of $\beta\log(11/10)$. 
The interpretation of $\beta$ is with regard to a doubling in $X$ (assuming the base of your log is 2): a doubling of the covariate from $X=x$ to $X=2x$ will change the expected value from $E[Y|X=x]=\alpha + \beta\log(x)$ to $E[Y|X=2x]=\alpha + \beta\log(2x) = \alpha + \beta + \beta\log(x)$, for a net change of $\beta$, and this holds for all values of $x$. 
You have presumably logged your covariate because you believe that changes in the covariate at small values will have a bigger impact than changes in the covariate at large values. For example, if you are modeling the association between income and some outcome related to financial anxiety, you probably would believe that a \$X increase in income is going to have a much different impact on financial anxiety if you're going from \$10K to \$10K + \$X than from \$300K to \$300K + \$X, even though the increase is the same. In that case, you might log income before fitting the model. If this logic doesn't apply to your circumstance, then you might reconsider your choice of logging the covariate. 
Finally, you note in your follow-up that you covariate can take on values from 0 to 365, inclusive. Your model cannot be interpreted when X=0 since $\log(0) = -\infty$. You could add one to the covariate prior to logging. 
