Problem: If (3x)(x2x)(2xy)=xx2\left(\log _{3} x\right)\left(\log _{x} 2 x\right)\left(\log _{2 x} y\right)=\log _{x} x^{2}(log3x)(logx2x)(log2xy)=logxx2, then yyy equals:
Answer Choices:
A. 9/29 / 29/2
B. 999
C. 181818
D. 272727
E. 818181 Solution:
Since ab=cbca\log _{a} b=\dfrac{\log _{c} b}{\log _{c} a}logab=logcalogcb, we have, with base xxx,
logxlog3log2xlogxlogylog2x=2\dfrac{\log x}{\log 3} \dfrac{\log 2 x}{\log x} \dfrac{\log y}{\log 2 x}=2 log3logxlogxlog2xlog2xlogy=2
logy=2log3=log9∴y=9\log y=2 \log 3=\log 9 \quad \therefore y=9 logy=2log3=log9∴y=9