zmin=10;
zmax=100;
d=5;
gridpoints=Subdivide[Log[zmin],Log[zmax],d]
breakpoints=Map[Exp,gridpoints]
fpoints=Map[Log,breakpoints]
Show[{Plot[Log[x],{x,0,110},Ticks->None],
Graphics[{Dashed,Map[Line,Thread[{Thread[{breakpoints,ConstantArray[0,d+1]}],
Thread[{breakpoints,fpoints}]}]]}],
Graphics[{Dashed,Map[Line,Thread[{Thread[{ConstantArray[0,d+1],gridpoints}],
Thread[{breakpoints,fpoints}]}]]}]}]