Question

The number of the solution is the equation $2x^{{\log }_{10}x}+3x^{{\log }_{10}\frac{1}{x}}=5$

• A. $1$
• B. $2$
• C. $3$
• D. None of these

Answer 1

The correct option is C $3$

Put ${\log }_{10}x=t$
$\therefore 2x^t+3x^{-t}=5$
$2(x^t{)}^2-5x^t$ $+3=0$
$\therefore (x^t-1)(2x^t-3)=0$
$\therefore x^t=1or\frac{3}{2}$
$x^t=1=x^0\Rightarrow t=0$
or ${\log }_{10}x=0$ $\therefore$ $x=1$
$x^t=\frac{3}{2}\Rightarrow t={\log }_x\frac{3}{2}$
${\log }_{10}x={\log }_x\frac{3}{2}={\log }_{10}\frac{3}{2}.{\log }_{x}10$
$(\because {\log }_{b}a={\log }_{c}a.{\log }_b=c)$
or $({\log }_{10}x{)}^2={\log }_{10}\frac{3}{2}$
$\therefore {\log }_{10}x=\pm \sqrt{{\log }_{10}\frac{3}{2}}$
$\therefore x={10}^{\pm \sqrt{{\log }_{10}3/2}}$
Hence only 3 solutions by (1) and (2).