In[]:=
(*thisexpressionshouldbezero,asthetwotermsareequal.Yes,IoverdidtheAssumptionsinanefforttomakeitwork.*)FullSimplify
-αt+4T-
2Log[-δλT]
ρ-α

-
-α(t+4T)

α
2Log[-δλT]
ρ-α

,Assumptions->{ρ>α>0,β>0,γ>0,δ>0,M>0,wT>0,λT<-1/δ<0,0<t<T,ρ∈Reals,α∈Reals}
Out[]=
-αt+4T+
2Log[-δλT]
α-ρ

-
-((t+4T)α)

2α
-α+ρ
(-δλT)
In[]:=
(*However,verysmallsimplificationsleadtothedesiredoutcomes.Here,Iremovethedenominator.*)​​FullSimplify[
-α(t+4T-2Log[-δλT])

-
-α(t+4T)

α(2Log[-δλT])

,Assumptions->{ρ>α>0,β>0,γ>0,δ>0,M>0,wT>0,λT<-1/δ<0,0<t<T,ρ∈Reals,α∈Reals}]
Out[]=
0
In[]:=
(*Also,Icanremoveonetermfromtheexponent:*)​​FullSimplify
-α4T-
2Log[-δλT]
ρ-α

-
-α(4T)

α
2Log[-δλT]
ρ-α

,Assumptions->{ρ>α>0,β>0,γ>0,δ>0,M>0,wT>0,λT<-1/δ<0,0<t<T,ρ∈Reals,α∈Reals}
Out[]=
0
(*Istheresomelimitonthenumberoftermsallowedinexponents.Itriedreplacingtwithaconstant,butstillgotthewrongresult.*)