Question

Find the value of
(i) $lo{g}_{1/2}8$
(ii) $lo{g}_{5}0.008$
(iii) $lo{g}_{5}8$
(iv) $lo{g}_7\sqrt[3]{37}$

Answer 1

Consider the given function:

$part\left(1\right){\log }_{\frac{1}{2}}8$

${\frac{1}{2}}^x=8where(\frac{1}{x^{-m}}=x^m)$

$\therefore x=-3$

${\left(\frac{1}{3}\right)}^{-3}=27$

$part\left(2\right){\log }_{5}0.008$

$let{\log }_5\mathrm{0.008......}bydefinationof\log arithm$

$\therefore 5^x=\frac{8}{1000}$

$\therefore 5^x=\frac{1}{125}=\frac{1}{5^3}$

$5^x=5^{-3}$

$x=-3$

$part\left(3\right){\log }_{5}8$

${\log }_{a}x=\frac{{\log }_{b}x}{{\log }_{b}a}$

$=\frac{\log 8}{\log 5}=\frac{0.9030}{0.6989}$

$=1.2920$

$part\left(4\right){\log }_7\sqrt[3]{37}$

$={\log }_7{7}^{\frac{1}{3}}where({\log }_b{b}^x=x)$

$=\frac{1}{3}$