How to log-linearize different forms of technology shocks?

Hello everyone, there are some question I would like to ask for your help. When I read paper, I find there are mainly two forms of shocks, take the technoloty shock as an example:
One kind of the forms is (1):1 , and for this form, the corresponding log-linearization is2 , and the steady state value of technology shock is 3 when in equilibrium. The other form of technology shock is (2):4 , for this form, I don’t know how to log-linearize the technology shock, is the same as the first form, and the steady state value of technology can be different from 1?
Besides, another question I would like to ask everyone, I want to construct a DSGE model incorporates two different producers, one producer with higher technology progress, the other with lower technology progress, that means for the first producer, its technology shock is 5 , and for the other producer, its technology shock progress is as follows: 6 . When in equilibrium,7 , and 8 , is this construction right? And how to realize this construction in DSGE model?
Looking forward for your reply, thank you very much!


the second expression is already log-linearized, why to do it a second time !!
And it is equivalent to the first expression you have just to set log A_{t}- logA= \hat{A}_{t}

Thank you for your reply~Here is another question, in the first form, the steady state value of A is 1 when in equilibrium, and only this value can hold in equilibrium, however in the second form, can I set the different steady state value which is different from 1? for example, A=2, because this value also hold in equilibrium.
Looking forward for your reply, thanks a lot!

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Exactly. Whenever you want to specify a steady state that is not equal to 0=log A \Leftrightarrow A=1 you need to use the second one.

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