Question

The product of all values of $x$ which make the following statement true $\left({\log }_{3}x\right)\left(lo{g}_{5}9\right)-{\log }_{x}25+{\log }_{3}2={\log }_{3}54$, is

• A. $\sqrt{5}$
• B. $5$
• C. $5\sqrt{5}$
• D. $25$

Answer 1

Answer: (C)

$5\sqrt{5}$

$\frac{\log x}{\log 3}.\frac{\log 9}{\log 5}-\frac{\log 25}{\log x}+\frac{\log 2}{\log 3}=\frac{\log 54}{\log 3}$

$\frac{\log x}{\log 3}.\frac{2\log 3}{\log 5}-\frac{2\log 5}{\log x}={\log }_3\left(\frac{54}{2}\right)={\log }_{3}27$

$2{\log }_{5}x-2\frac{1}{{\log }_{5}x}=3$

$let{\log }_{5}x=z$

$2z-\frac{2}{z}=3$

$2(z^2-1)=3z$

$2z^2-2-3z=0$

$2z^2-3z-2=0$

$2z^2-4z+z-2=0$

$2z(z-2)+1(z-2)=0$

$(2z+1)(z-2)=0$

$z=\frac{-1}{2}andz=2$

${\log }_{5}x=\frac{-1}{2}and{\log }_{5}x=2$

$x=5^{\frac{-1}{2}}andx=5^2$

$let{x}_1=5^{\frac{-1}{2}}and{x}_2=5^2$

$x_1{x}_2=5^2\times 5^{\frac{-1}{2}}=5^{\frac{3}{2}}=5\sqrt{5}$