Marshallian demand with Leontif preferences Consider a utility function on the form $u(q_{1},q_{2},q_{3}) = min\{\alpha ln(q_{1}) + (1 - \alpha) ln(q_{2}), ln(q_{3})\}$

I know that optimal behaviour requires $\alpha ln (q_{1}) + (1 - \alpha) ln(q_{2}) = ln(q_{3})$

I tried substituting this into the budget constraint (after raising both sides to the power of $e$)

$p_{1}q_{1} + p_{2}q_{2} = y$

But I am unsure how to proceed in finding the Marshallian demand, since I have two variables $(q_{1}, q_{2})$ and only one equation. I have tried different transformations but my main problem is that it feels like I'm missing an equation.

Any help is greatly appreciated. 

EDIT: I just re-read the question and I misspecified the budget constraint! The consumer only spends on $q_{1}$ and $q_{2}$.
So, I have not really worked out the maths behind solving this. I prefer to take a shortcut and just compute it numerically. Here is what I have. 

These are the contour surfaces and the budget constraint. Notice how the solution will be at the "kinked" curve in the middle of each surface. 


The levels 1 and 2 are only for example. The third level, u = 3.92762 is related to the next part of my answer. Also, BC is the budget curve with income = 300. I have chosen $\alpha = 0.5$ and $P_1 = 2, P_2=3, P_3=1$. 

Now, using the parameters mentioned in the last paragraph, the maximum is found to be at $q_1 = 65.8896, q_2 = 39.1448, q_3 = 50.7863$ and the maximized utility is $u = 3.92762$. I assumed that all goods are strictly consumed at positive amounts. 

I am attaching my Mathematica code below, so that you can replicate and modify these results as you want


  u = Min[a*Log[q1] + ((1 - a)*Log[q2]), Log[q3]]   
  
  BC = p1*q1 + p2*q2 + p3*q3
    a = 0.5; p1 = 2; p2 = 3; p3 = 1;
  
  ContourPlot3D[{u == 1, u == 2, u == 3.92762, BC == 300}, {q1, 0,
  100}, {q2, 0, 100} , {q3, 0, 100}, AxesLabel -> Automatic,
  ContourStyle -> {Opacity[0.5], Opacity[0.5], Opacity[0.5],     Opacity
  [1]}, Mesh -> None, PlotLegends -> "Expressions"]
  
  NMaximize[{u, BC &lt;= 300, q1 > 0, q2 > 0, q3 > 0}, {q1, q2, q3}, Reals]