Question

Solve the following equations.

$x^{3lo{g}^{3}x-\frac{2}{3}logx}=100\sqrt[3]{310}$

Answer 1

$x^{3lo{g}^{3}x-\frac{2}{3}logx}=100(10{)}^{1/3}$

$\Rightarrow x^{3lo{g}^{3}x-\frac{2}{3}logx}={10}^{2+1/3}$

Taking logarithm an both sides (base will be to)

$\Rightarrow (3lo{g}^{3}x-\frac{2}{3}logx)logx=\frac{7}{3}log10$

$\Rightarrow 3lo{g}^{4}x-\frac{2}{8}lo{g}^{2}x=\frac{7}{3}$

Say $lo{g}^{2}x=t$

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

$\Rightarrow 9t^2-2t=7$

$\Rightarrow 9t^2-9t+7t-7=0$

$\Rightarrow 9t(t-1)+7(t-1)=0$

$\therefore t=1$ or $t=-7/9$

$\Rightarrow lo{g}^{2}x=1\Rightarrow lo{g}^{2}x=-7/9$

Now we know that the square of a quantity cannot be negative

$\therefore$ the only solution is

$logx=1$

$x=10$