Question

Solve the following equations.
(i) ${\log }_{2}x=\frac{1}{2}$
(ii) ${\log }_{\frac{1}{5}}x=3$
(iii) ${\log }_{3}y=-2$
(iv) ${\log }_{x}125\sqrt{5}=7$
(v) ${\log }_{x}0.001=-3$
(vi) $x+2{\log }_{27}9=0$

Answer 1

(i) $x=2^{1/2}=\sqrt{2}$

(ii) $x={\left(\frac{1}{5}\right)}^3=\frac{1}{125}$

(iii) $y=3^{-2}=\frac{1}{9}$

(iv) $x^7=125\sqrt{5}=(\sqrt{5}{)}^7\Rightarrow x=\sqrt{5}$

(v) $0.001=x^{-3}\Rightarrow x=10$

(vi) $x+2{\log }_{3^3}{3}^2=0\Rightarrow x=-2\times \frac{2}{3}{\log }_{3}3\Rightarrow x=\frac{-4}{3}$