In[]:=
Resolve[ForAll[{x,y,z},Implies[x>0&&y>0&&z>0&&2^x==3^y==5^z,#]],Reals]&/@{2x<3y<5z,5z<2x<3y,3y<5z<2x,3y<2x<5z}
Out[]=
{False,False,False,True}
In[]:=
Reduce[{2^x==3^y==5^z,x>0,#},{x,y,z},Reals]&/@{2x<3y<5z,5z<2x<3y,3y<5z<2x,3y<2x<5z}
Out[]=
False,False,False,x>0&&y
xLog[2]
Log[3]
&&z
yLog[3]
Log[5]
