This is not comprehensible (and the subject title is meaningless). What exactly is wanted? Something like this?
In[27]:= phi = 0.5; kap = 0.2; u =
RandomReal[{0, 1},
10]; x = (1/phi)*
Log[(kap - 1 + ((1 + kap)^2 - 4 kap u)^0.5)/(2 kap)] - (1/
phi^2) - (1/phi)
Out[27]= {-7.50006616149, -6.17473673094, -9.36459249618, \
-8.78032746892, -9.33316651726, -9.80992590354, -6.29708140172, \
-6.01196696005, -6.34504903014, -6.70520204079}
In[28]:= ProductLog[(1/phi)*
Exp[-(1/phi)] ((kap - 1 + ((1 + kap)^2 - 4 kap u)^0.5)/(2*kap))]
Out[28]= {0.114068677509, 0.202549549343, 0.0479727646523, \
0.063273418175, 0.0486972217016, 0.0387521917698, 0.192462165763, \
0.216647476984, 0.188623734815, 0.161819558864}