One way is using NMaximize:
NMaximize[x1*Exp[Log[600] - (4/600)*x1] + x2*Exp[Log[340] - (1/1020)*x2], {x1, x2}]
(* {160690., {x1 -> 150., x2 -> 1020.}} *)
or:
Maximize[x1*Exp[Log[600.] - (4./600.)*x1] + x2*Exp[Log[340.] - (1/1020.)*x2], {x1, x2}]
(* {160690., {x1 -> 150., x2 -> 1020.}} *)
FindMaximum[x1*Exp[Log[600] - (4/600)*x1] + x2*Exp[Log[340] - (1/1020)*x2], {x1, x2}]
(* {160690., {x1 -> 150., x2 -> 1020.}} *)