Mathematica solves the equation and gives the empty set, which is correct, as you can check by turning the system into a linear one:
eqs = {7463^x*2070^y*15^w*10^z == 3,
7060^x*2035^y*15^w*10^z == 5,
8874^x*1945^y*15^w*10^z == 23,
9918^x*1949^y*15^w*10^z == 17};
toLog = Map[Log, eqs, {2}] /.
HoldPattern[Log[Times[x__]]] :> Plus @@ Map[Log, {x}] /.
Log[a_^b_] :> b*Log[a]
{b, mat} = CoefficientArrays[toLog, {w, x, y, z}] // Normal
mat // Simplify // MatrixForm
Det[mat]
