Question

Prove the following equations.
(i) ${\log }_{10}1600=2+4{\log }_{10}2$
(ii) ${\log }_{10}12500=2+3{\log }_{10}5$
(iii) ${\log }_{10}2500=4-2{\log }_{10}2$
(iv) ${\log }_{5}0.00125=3-5{\log }_{5}10$

Answer 1

(i) ${\log }_{10}1600={\log }_{10}100+{\log }_{10}16=2+{\log }_{10}{2}^4=2+4{\log }_{10}2$

(ii) ${\log }_{10}12500={\log }_{10}100+{\log }_{10}{5}^3=2+3{\log }_{10}5$

(iii) ${\log }_{10}2500={\log }_{10}\frac{10000}{4}=4-{\log }_{10}4=4-2{\log }_{10}2$

(iv) ${\log }_{5}0.00125={\log }_5\left(\frac{125}{{10}^5}\right)={\log }_5{5}^3-{\log }_5{10}^5=3-5{\log }_{5}10$