cm=0.01;
eqn=100==10(cm^2)*(5*(T-20)+0.8*5.6*(10^(-8))*((((T+273))^4)-(293^4)));

Log[-1]

res=Solve[eqn]

Cases[res,{T->_?((Im[#]==0&&Re[#]>0)&)},{0,∞}]

And

T/.res

Select[{-1519.1272230387428`,-254.48995999030527`-1227.8962461179267` I,-254.48995999030527`+1227.8962461179267` I,936.1071430193532`},#>0&]

Quiet@Select[T/.res,#>0&]

((Im[#]==0&&Re[#]>0)&)[3]

Cases[res,{T->_?((Im[#]==0&&Re[#]>0)&)},{0,∞}]

list=Range[10]
test[x_]:=x>5;
Cases[list,_?test]

FindRoot[eqn,{T,50}]

Cases[res,{T->_?((Im[#]==0&&Re[#]>0)&)},{0,∞}]

eqna=100==10(cm^2)*(5*(T-20)+0.8*5.6*(10^(-8))*(Ψ-(293^4)));
ClearAll[Tnext]
Tnext[Tprevious_]:=((Ψ/.Solve[eqna,Ψ][[1]]/.T->Tprevious)/.Ψ->(((Tprevious+273))^4))^(1/4)-273;
NestList[Tnext,1000,5]

eqnb=100==10(cm^2)*(5*(Ψ-20)+0.8*5.6*(10^(-8))*((((T+273))^4)-(293^4)));
ClearAll[Tnext]
Tnext[Tprevious_]:=((Ψ/.Solve[eqnb,Ψ][[1]]/.T->Tprevious));
NestList[Tnext,1000,5]

rhs=eqn[[2]];
lhs=eqn[[1]];
Tnext[Tprevious_]:=Module[{ser},
ser=Series[rhs,{T,Tprevious,1}]//Normal;
T/.Solve[ser==lhs][[1]]
];
NestList[Tnext,1000,5]

(*load flow problem, cartesian form of complex numbers*)
Quiet@Remove["Global`*"];
simplify=#&;(*"lower" programing languages have no Mathematica Simplify[#]&*)
(*simplify=Simplify;*)
conjugate[{re_,im_}]:={re,-im};
times[ua:{uare_,uaim_},ub:{ubre_,ubim_}]:={uare*ubre-uaim*ubim,uare*ubim+ubre*uaim};
re[{re_,im_}]:={re,0};
im[{re_,im_}]:={im,0};
sqabs[{re_,im_}]:={(re^2)+(im^2),0};
powerMinus1[z:{re_,im_}]:=(({re,-im})/((re^2)+(im^2)));
ratio[a_,b_]:=times[powerMinus1[b],a]
apparentPower[u_,i_]:=times[u,conjugate[i]];
activePower[u_,i_]:=re[apparentPower[u,i]];
reactivePower[u_,i_]:=im[apparentPower[u,i]];
current[u1_,u2_,z_]:=ratio[u1-u2,z];
toMyForm[z_]:={Re[z],Im[z]};
fromMyForm[{re_,im_}]:=re+I*im;
rearrangeRealEqn[eqn_Equal]:={simplify[eqn]}/.Equal[a_,b_]:>Equal[a-b,0];
rearrangeComplexEqn[{rea_,ima_}=={reb_,imb_}]:=DeleteCases[#,rearrangeRealEqn[True]]&@Flatten[rearrangeRealEqn/@{simplify[rea==reb],simplify[ima==imb]}];

kz=2.5;(*respects length of lines*)
kReg=2.5;(*ratio of powers of sources*)
{(Subscript[z, 1]),(Subscript[z, 2]),(Subscript[z, 3]),(Subscript[z, 4])}=kz*(toMyForm/@{.01+0.05I,2*(01+0.05I),0.01+0.02I,0.01+0.02I});
powerFactor4=0.8;
powerFactor5=0.7;
S1=(10^3)*{powerFactor4,Sin[ArcCos[powerFactor4]]};
S2=500*{powerFactor5,Sin[ArcCos[powerFactor5]]};
{(Subscript[u, #])={(Subscript[ure, #]),(Subscript[uim, #])},(Subscript[i, #])={(Subscript[ire, #]),(Subscript[iim, #])}}&/@Range[6];
Unom=230.;

eqnZ1=rearrangeComplexEqn[(Subscript[i, 1])==current[(Subscript[u, 1]),(Subscript[u, 2]),(Subscript[z, 1])]];
eqnZ2=rearrangeComplexEqn[(Subscript[i, 2])==current[(Subscript[u, 2]),(Subscript[u, 3]),(Subscript[z, 2])]];
eqnZ3=rearrangeComplexEqn[(Subscript[i, 2])==current[(Subscript[u, 5]),(Subscript[u, 3]),(Subscript[z, 3])]];
eqnZ4=rearrangeComplexEqn[(Subscript[i, 2])==current[(Subscript[u, 4]),(Subscript[u, 3]),(Subscript[z, 4])]];
eqnApparentPower1=rearrangeComplexEqn[apparentPower[(Subscript[u, 2]),(Subscript[i, 5])]==S1];
eqnApparentPower2=rearrangeComplexEqn[apparentPower[(Subscript[u, 3]),(Subscript[i, 6])]==S2];
eqnNod1=rearrangeComplexEqn[(Subscript[i, 1])==(Subscript[i, 2])+(Subscript[i, 5])];
eqnNod2=rearrangeComplexEqn[(Subscript[i, 6])==(Subscript[i, 2])+(Subscript[i, 3])+(Subscript[i, 4])];
eqnVoltage2=rearrangeComplexEqn[sqabs[(Subscript[u, 2])]==toMyForm[(Unom^2)]];
eqnVoltage3=rearrangeComplexEqn[sqabs[(Subscript[u, 3])]==toMyForm[(Unom^2)]];
eqnSourceActivePowerRatio1=rearrangeComplexEqn[activePower[(Subscript[u, 1]),(Subscript[i, 1])]==2*activePower[(Subscript[u, 5]),(Subscript[i, 3])]];
eqnSourceActivePowerRatio2=rearrangeComplexEqn[reactivePower[(Subscript[u, 1]),(Subscript[i, 1])]==3*reactivePower[(Subscript[u, 4]),(Subscript[i, 4])]];
eqnSourceReactivePowerRatio1=rearrangeComplexEqn[reactivePower[(Subscript[u, 1]),(Subscript[i, 1])]==2*reactivePower[(Subscript[u, 5]),(Subscript[i, 3])]];
eqnTime={(Subscript[ure, 3])==0};
eqnsTogether=Union[eqnZ1,eqnZ2,eqnZ3,eqnZ4,eqnApparentPower1,eqnApparentPower2,eqnNod1,eqnNod2,eqnVoltage2,eqnVoltage3,eqnSourceActivePowerRatio1,eqnSourceActivePowerRatio2,eqnSourceReactivePowerRatio1,eqnTime];
unknownVariables=Union@Cases[eqnsTogether,Subscript[_,_],{0,∞}];
Length/@{eqnsTogether,unknownVariables}

vectfces=First/@eqnsTogether;
vars=unknownVariables;
grad[expr_]:=D[expr,#]&/@vars;
jacobian=grad/@vectfces;
mf@%;
xStart=unknownVariables/.{(Subscript[iim, index_]):>1,(Subscript[ire, index_]):>1,(Subscript[ure, index_]):>Unom,(Subscript[uim, index_]):>Unom};
xNext[xPrevious_]:=xPrevious-Inverse[jacobian/.Thread[vars->xPrevious]].(vectfces/.Thread[vars->xPrevious]);
aux=NestList[xNext,xStart,50][[-1]]

mf=MatrixForm;
vectfces=First/@eqnsTogether;
vars=unknownVariables;
grad[expr_]:=D[expr,#]&/@vars;
jacobian=grad/@vectfces;
mf@%;
xStart=unknownVariables/.{(Subscript[iim, index_]):>1,(Subscript[ire, index_]):>1,(Subscript[ure, index_]):>Unom,(Subscript[uim, index_]):>Unom};
xNext[xPrevious_]:=xPrevious-Inverse[jacobian/.Thread[vars->xPrevious]].(vectfces/.Thread[vars->xPrevious]);
aux=NestList[xNext,xStart,50][[-1]];