If 2^x=3^y=12^z then show that 1/z = 1/y + 2/x

Answer 1

take ln of each side:

xln2 = yln3 = zln12

z = yln3/ln12 , and y = xln2/ln3

so z =(xln2/ln3)(ln3/ln12) = xln2/ln12

LS = 1/z = ln12/(xln2)
RS = 1/y + 2/x
= ln3/(xln2) + 2/x
= (ln3 + 2ln2)/(xln2)
= (ln3 + ln(2^2))/(xln2)
= (ln3 + ln4)/(xln2)
= ln12/(xln2)
= LS