I am analyzing ARIMA or multiple regression model. $Y= X_1 + X_2 + X_3 + X_4$
Can I have the log on Y and square root on X1 and square root on X2, at the same time? That is,
$\log Y = \sqrt{X_1} + \sqrt{X_2} + X_3 + X_4$
If yes, how can I interpret the results?
Yes; you can do this and still achieve a valid model.
Interpretation, however, is complicated by this transformation. Your model then looks like
$$\log(E(Y_i)) = \beta_0+ \beta_1\sqrt{X_{i1}} + \beta_2\sqrt{X_{i2}} + \beta_3X_{i3} + \beta_4X_{i4}$$
This means that e.g. a doubling of $X_1$ is associated with an expected increase of $Y$ by a factor $\exp(\beta_1\sqrt{2})$, all other factors kept equal. For a unit increase in $X_1$ there is no easy interpretation; that depends on the current value of $X_1$.
A unit increase of $X_3$ is associated with an expected increase of $Y$ by a factor $\exp(\beta_3)$. So, interpretation is easier without transforming the variables. However, if square root transformation improves the fit or is necessary in some other way, then it is mathematically perfectly correct.
