Question

If $5x^{lo{g}_{2}3}+3^{lo{g}_{2}x}=162$ then logarithm of $x$ to the base 4 has the value equal to :

• A. $2$
• B. $1$
• C. $-1$
• D. $3/2$

Answer 1

The correct option is D $3/2$

Given,

$5x^{{\log }_2\left(3\right)}+3^{{\log }_2\left(x\right)}=162$

$3^{{\log }_2\left(x\right)}=162-5x^{{\log }_2\left(3\right)}$

$\Rightarrow {\log }_2\left(3^{{\log }_2\left(x\right)}\right)={\log }_2(162-5x^{{\log }_2\left(3\right)})$

$\Rightarrow {\log }_2\left(x\right){\log }_2\left(3\right)={\log }_2(162-5x^{{\log }_2\left(3\right)})$

$\Rightarrow {\log }_2\left(x^{{\log }_2\left(3\right)}\right)={\log }_2(162-5x^{{\log }_2\left(3\right)})$

$x^{{\log }_2\left(3\right)}=162-5x^{{\log }_2\left(3\right)}$

$x={27}^{\frac{665}{1054}}\approx 8$

Now,

${\log }_{4}x$

$={\log }_{4}8$

$=\frac{3}{2}$