This paper is concerned with the following logarithmic Schr\"{o}dinger
system: ⎩⎨⎧−Δu1+ω1u1=μ1u1logu12+p+q2p∣u2∣q∣u1∣p−2u1,−Δu2+ω2u2=μ2u2logu22+p+q2q∣u1∣p∣u2∣q−2u2,∫Ω∣ui∣2dx=ρi,i=1,2,(u1,u2)∈H01(Ω;R2), where Ω=RN or
Ω⊂RN(N≥3) is a bounded smooth domain,
ωi∈R, μi,ρi>0,i=1,2. Moreover, $p,\ q\geq1,\
2\leq p+q\leqslant 2^*,where2^*:=\frac{2N}{N-2}.ByusingaGagliardo−Nirenberginequalityandcarefulestimationofu\log u^2,firstly,wewillprovideaunifiedproofoftheexistenceofthenormalizedgroundstatessolutionforall2\leq p+q\leqslant 2^*.Secondly,weconsiderthestabilityofnormalizedgroundstatessolutions.Finally,weanalyzethebehaviorofsolutionsforSobolev−subcriticalcaseandpassthelimitastheexponentp+qapproachesto2^*.Notably,theuncertaintyofsignofu\log
u^2in(0,+\infty)isoneofthedifficultiesofthispaper,andalsooneofthemotivationsweareinterestedin.Inparticular,wecanestablishtheexistenceofpositivenormalizedgroundstatessolutionsfortheBreˊzis−Nirenbergtypeproblemwithlogarithmicperturbations(i.e.,p+q=2^*).Inaddition,ourstudyincludesprovingtheexistenceofsolutionstothelogarithmictypeBreˊzis−NirenbergproblemwithandwithouttheL^2−mass\int_{\Omega}|u_i|^2\,dx=\rho_i(i=1,2)$ constraint by two different
methods, respectively. Our results seems to be the first result of the
normalized solution of the coupled nonlinear Schr\"{o}dinger system with
logarithmic perturbation