Question

if 2^z=3^y=12^z, show that 1/z=1/y=2/x

Answer 1

Let 2x=3y=12z=r2x=3y=12z=r. Then

2=r1/x,3=r1/y2=r1/x,3=r1/y, and 12=r1/z12=r1/z.

Since 12=22⋅312=22⋅3,

r1/z=r2/x⋅r1/yr1/z=r2/x⋅r1/y,

or

2x=1z−1y=y−zyz2x=1z−1y=y−zyz.

Thus x=2yzy−zx=2yzy−z. ■◼