Question

If $x_1$ and $x_2$ are the solution of the equation $x^{3\log {}_{10}^{3}x-\frac{2}{3}{\log }_{10}x}=100\sqrt[3]{310}$, then which of the following is true?

• A. $x_1{x}_2=1$
• B. $x_1.x_2=x_1+x_2$
• C. ${\log }_{x_2}{x}_1=-1$
• D. $\log (x_1.x_2)=0$

Answer 1

Answer: (A), (C), (D)

$x_1{x}_2=1$
${\log }_{x_2}{x}_1=-1$
$\log (x_1.x_2)=0$

$x^{3\log {}_{10}^{3}x-\frac{2}{3}{\log }_{10}x}={10}^{\frac{7}{3}}$

Taking log both sides on base 10,

$\Rightarrow \left(\begin{array}{c}3{\log }_{10}^{3}x-\frac{2}{3}{\log }_{10}x\end{array}\right){\log }_{10}x=\frac{7}{3}[\because \log a^m=m\log a\text{\&}{\log }_{a}a=1]$

Substitute ${\log }_{10}x=t$

$\Rightarrow 3t^4-\frac{2}{3}{t}^2=\frac{7}{3}\Rightarrow 9t^4-2t^2-7=0$

$\Rightarrow (9t^2+7)(t^2-1)=0\therefore t=\pm 1$

$\Rightarrow {\log }_{10}x=\pm 1\Rightarrow x={10}^{\pm 1}$

$\Rightarrow x=10,\frac{1}{10}$$\therefore x_1.x_2=1,{\log }_{x_2}{x}_1=-1,\log (x_1.x_2)=0$

Ans: A,C,D