Question

Find the greatest value of $\ln (x) \ln (z)$ provides that $\ln (x)+\log _y(z)=3$ and $\ln (y)+\log _x(z)=$ 4 where $x, y, z>1$

Answer 1

$\text { Suppose } \ln (x)=a, \ln (y)=b, \ln (z)=c$
$\text { Since } x, y, z >1 $
$\text { Hence } a+(c / b)=3, b+(c / a)=4 $
$a b+c=3 b=4 a$
$\text { Suppose } a=3 u, b=4 u(u>0) \text { then } $
$c=4 a-a b=12 u-12 u^2$
$\ln x \ln z=a c=3 u .12 u(1-u)=18 u^2(u-2 u) \leq 18\left(\frac{u+u+(2-2 u)^3}{3}\right)=18\left(\frac{2}{3}\right)^3=\frac{16}{3}$