Question

Simplify (i) ${\log }_{5}25+{\log }_{5}625$
(ii) ${\log }_{5}4+{\log }_5\left(\frac{1}{100}\right)$

Answer 1

(i) ${\log }_{5}25+{\log }_{5}625={\log }_5(25\times 625)$ $[\because {\log }_a(M\times N)={\log }_{a}M+{\log }_{a}N]$
$={\log }_5(5^2\times 5^4)={\log }_5{5}^6=6{\log }_{5}5$ $[\because {\log }_a{\left(M\right)}^n=n{\log }_{a}M]$
$=6\left(1\right)=6$ $[\because {\log }_{a}a=1]$
(ii) ${\log }_{5}4+{\log }_5\left(\frac{1}{100}\right)={\log }_5(4\times \frac{1}{100})$ $[\because {\log }_a(M\times N)={\log }_{a}M+{\log }_{a}N]$
$={\log }_5\left(\frac{1}{25}\right)={\log }_5\left(\frac{1}{5^2}\right)={\log }_5{5}^{-2}=-2{\log }_{5}5$ $[\because {\log }_a{\left(M\right)}^n=n{\log }_{a}M]$
$=-2\left(1\right)=-2$ $[\because {\log }_{a}a=1]$
log525+log5625=log5(25×625)