Log linearization

I’m working on sentiment effect on aggregate demand and face a conditional expectation equation. How can I linearize the last equation?
Firms.pdf (35.5 KB)

What is the problem with this equation. You can use a standard first order Taylor approximation.

In our original equation, along with production, employment and idiosyncratic shocks at the firm level we also have the total price. Is it alright? and Should I consider a constant the variance?
log linearization.pdf (84.2 KB)

Every variable, i.e. everything that has a time index needs to be considered. If there is a price level in the original equation, it should appear in the final linearized equation as well. However, there are cases where only the real variables and inflation are uniquely determined as in most New Keynesian models (the price level has a unit root). In that case, it is advisable to linearize the real wage instead of the nominal wage and the price level separately.
Now regarding variances: they do not have a time index in your equations and are therefore constant. If you did a mistake here and the variances are time-varying, then of course this needs to be considered during linearization.

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My mean from the firm level was the variable with jt subscript and the total price was the variable with t subscript. Can be both placed in one equation?

That depends on your model and how you handle aggregation. Only you, the model-builder, can answer this.