GBM formulas – when to subtract 0.5σ^2 from u

Background – I often get confused when (not) to subtract. Here’s a brief summary.
The standard GBM dynamic is
                dS = mu S dt + σ S dW …. where mu and σ are time-invariant.
The standard solution is to find the dynamics of logS, denoted L,
                dL = (mu – 0.5σ2 ) dt + σ dW …  BM, not GBM. No L on the RHS.
               
                L (time=T)     ~  N (mean = (mu – 0.5σ2 )T, std = …. )
So it seems our mu can’t get rid of –0.5σ2 thingy … Until we take expectation of S(time=T)
                E S(time=T) = S(time=0) exp(mu*T)     … no σ2 term
When we write down the Black Scholes PDE we use mu without the –0.5σ2 thingy.
BS formula uses mu without the –0.5σ2 thingy.

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