Solving simultaneous equations gives no output for long time

Questions
1

I want to solve this system of equations in terms of [Nu]1 and [Nu]2, but Mathematica has been running for a day without returning any output. I have simplified the equations as much as I could, and have tried using Solve as well. Is there any way to make this quicker?

(*Define symbols*)
m11 = Subscript[m, 11] = Sqrt[2300000];
m22 = Subscript[m, 22] = Sqrt[280000];
\[Lambda]1 = Subscript[\[Lambda], 1] = 0.1305;
\[Lambda]2 = Subscript[\[Lambda], 2] = 0.842;
\[Lambda]345 = Subscript[\[Lambda], 345] = 0;
\[Nu]1 = Subscript[\[Nu], 1];
\[Nu]2 = Subscript[\[Nu], 2];

(*Define A and B*)
A = 2 m11^2 + 2 m22^2 + 
   3 \[Lambda]1 \[Nu]1^2 + \[Lambda]345 \[Nu]1^2 + 
   3 \[Lambda]2 \[Nu]2^2 + \[Lambda]345 \[Nu]2^2;
B = 3 \[Lambda]1 \[Lambda]345 \[Nu]1^4 - 
   3 \[Lambda]345^2 \[Nu]1^2 \[Nu]2^2 + 
   9 \[Lambda]1 \[Lambda]2 \[Nu]2^4 + 
   3 \[Lambda]2 \[Lambda]345 \[Nu]2^4 + 6 \[Lambda]1 m22^2 \[Nu]1^2 + 
   2 \[Lambda]345 m11^2 \[Nu]1^2 + 6 \[Lambda]2 m11^2 \[Nu]2^2 + 
   2 \[Lambda]345 m22^2 \[Nu]2^2 + 4 m11^2 m22^2;

(*Define E' and H'*)
Eprime = 
  2 \[Nu]1 (6 \[Lambda]1 \[Lambda]345 \[Nu]1^2 - 
     3 \[Lambda]345^2 \[Nu]2^2 + 9 \[Lambda]1 \[Lambda]2 \[Nu]2^2 + 
     6 \[Lambda]1 m22^2 + 2 \[Lambda]345 m11^2);
Hprime = 
  2 \[Nu]2 (6 \[Lambda]2 \[Lambda]345 \[Nu]2^2 - 
     3 \[Lambda]345^2 \[Nu]1^2 + 9 \[Lambda]1 \[Lambda]2 \[Nu]1^2 + 
     6 \[Lambda]2 m11^2 + 2 \[Lambda]345 m22^2);

(*Write the equations*)
eq1 = -Log[B] + Log[4] + 4 Log[\[Mu]] - 
    2 == (2 A/Sqrt[A^2 - 4 B])*(Log[A + Sqrt[A^2 - 4 B]] - Log[4] - 
      2 Log[\[Mu]]);
eq2 = Eprime == Hprime;


(*Display and solve the equations*)
Simplify[eq1]
Simplify[eq2]

Solve[{eq1, eq2}, {\[Nu]1, \[Nu]2}]

Edit: The question has been edited to include the parameter values. Now, Mathematica completes the evaluation but returns no output.

2

It is a transcendental system with six parameters, pretty hopeless to solve symbolically. You can try giving the parameters some values:

paramValues = {m11 -> 1, m22 -> 2,
   \[Lambda]1 -> 3, \[Lambda]2 -> -2,
   \[Lambda]345 -> 1, \[Mu] -> 4};
ContourPlot[Evaluate[{eq1, eq2} /. paramValues],
 {Subscript[\[Nu], 1], 0, 5}, {Subscript[\[Nu], 2], -3, 0}]
FindRoot[{eq1, eq2} /. paramValues,
 {{Subscript[\[Nu], 1], 2}, {Subscript[\[Nu], 2], -1}}]