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, z1

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 (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

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 u2 ln x ln z=a c=3 u .12 u(1-u)=18 u2(u-2 u) ≤ 18((u+u+(2-2 u)3/3))=18((2/3))3=(16/3)

Q. Find the greatest value of ln(x)ln(z) provides that ln(x)+logy​(z)=3 and ln(y)+logx​(z)= 4 where x,y,z>1

Suppose ln(x)=a,ln(y)=b,ln(z)=c Since x,y,z>1 Hence a+(c/b)=3,b+(c/a)=4 ab+c=3b=4a Suppose a=3u,b=4u(u>0) then c=4a−ab=12u−12u2 lnxlnz=ac=3u.12u(1−u)=18u2(u−2u)≤18(3u+u+(2−2u)3​)=18(32​)3=316​

Q. Find the greatest value of $ln (x) ln (z)$ provides that $ln (x) + lo g_{y} (z) = 3$ and $ln (y) + lo g_{x} (z) =$ 4 where $x, y, z > 1$