Λ[ϵ_, b_, c_,u_,w_,α_,β_,δ_]:=With[{ν=(((1+b δ))^(-1))},Simplify[ϵ(2 c w α δ ν+2 c u β δ ν+2 c u w (δ^2) ν+u (w^2) α (δ^2) (ν^3)-(1/2) b (u^2) (w^2) (δ^4) (ν^3)+(1/2) (w^2) (α^2) δ (2+b δ) (ν^3)-(u^2) w β (δ^2) (1+2 b δ) (ν^3)-(1/2) (u^2) (β^2) δ (2+3 b δ) (ν^3)+2 c (δ+α β ν)-2 b u w (δ^2) (ν^2) (δ+α β ν)+w α (ν^2) (2 δ+α β ν)-u β (1+2 b δ) (ν^2) (2 δ+α β ν)-(1/2) b ν (2 (δ^2)+4 α β δ ν+(α^2) (β^2) (ν^2)))]
]

logos=Λ[1, b, c,u,w,α,β,δ]

Solve[ν==(((1+b δ))^(-1)),b]

(Collect[Simplify[logos /.b->((1-ν)/(δ ν))], {c,u,w}, C[Simplify[#]]&]//Expand)/.C[x_]:>x

(Collect[Simplify[(((1+b δ))^(-1))logos /.b->((1-ν)/(δ ν))], {c,u,w}, C[Simplify[#]]&]//Expand)/.C[x_]:>x

ncore=With[{ζ=(b/(t-1))}, -b ν α β+ν β u+ν δ u w+ν α w +Θ /. {β->(ζ^(-1))β, δ->(ζ^(-1))δ, ν->(((1+b δ))^(-1))} /.  b->((1-ν)/(δ ν))] // Simplify

nlogos=With[{ζ=(b/(t-1))}, Λ[ζ ϵ,b,c,ζ u,w,α,(ζ^(-1))β,(ζ^(-1))δ]]

FreeQ[nlogos,b]

(Collect[(ϵ^(-1))Simplify[nlogos], {c,u,w}, C[Simplify[#]]&]//Expand)/.C[x_]:>x

(Collect[Simplify[(ϵ^(-1))2 (((1+(-1+t) δ))^3)nlogos], {c,u,w}, C[Simplify[#]]&]//Expand)/.C[x_]:>x

Solve[(((1+(-1+t) δ))^(-1))==μ,t]

(Collect[Simplify[(ϵ^(-1))nlogos /. t->((1-μ+δ μ)/(δ μ))], {c,u,w}, C[Simplify[#]]&]//Expand)/.C[x_]:>x

Simplify[(((1+(-1+t) δ))^3)nlogos/ϵ /. {t->(Subscript[t, k]),c->(Subscript[c, k]),u->(Subscript[u, k]),w->(Subscript[w, k])}]

(Collect[Simplify[(((1+(-1+t) δ))^3)nlogos/ϵ ], {c,u,w}, C[Simplify[#]]&]//Expand)/.{C[x_]:>x, t->(Subscript[t, k]),c->(Subscript[c, k]),u->(Subscript[u, k]),w->(Subscript[w, k])}

With[{ζ=(b/(t-1))}, Λ[ϵ,b,c,ζ u,w,α,(ζ^(-1))β,(ζ^(-1))δ]]