Question

If ${\log }_{10}8=0.90;$ find the value of :

a)${\log }_{10}4$

b)$\log \sqrt{32}$

c)$\log 0.125$

Answer 1

${\log }_{10}8=0.9=3{\log }_{10}2=0.9$

$={\log }_{10}2=0.3$

$\left(i\right){\log }_{10}4={\log }_{10}{2}^2=2{\log }_{10}2=2\times 0.3=0.6$

$\left(ii\right){\log }_{10}\sqrt{32}=\frac{1}{2}{\log }_{10}32=\frac{1}{2}{\log }_{10}{2}^5=\frac{5}{2}{\log }_{10}2=\frac{5}{2}\times 0.3=0.75$

$\left(iii\right){\log }_{10}0.125={\log }_{10}125\times {10}^{-3}={\log }_{10}{5}^3+{\log }_{10}{10}^{-3}=3{\log }_{10}5-3$

$=-3+3{\log }_{10}\left(\frac{10}{2}\right)=-3+3[{\log }_{10}10-{\log }_{10}2]=-3{\log }_{10}2=-0.9$