Tính giá trị của các biểu thức sau: a) log 3 9/10 + log 3 30

Trả lời

a) ${\log }_3\frac{9}{10}+{\log }_{3}30={\log }_3\left(\frac{9}{10}.30\right)={\log }_3{3}^3=3$ ;

b) ${\log }_{5}75-{\log }_{5}3$$={\log }_5\frac{75}{3}={\log }_{5}25$$={\log }_5{5}^2=2$ ;

c)  ${\log }_3\frac{5}{9}-2{\log }_3\sqrt{5}$$={\log }_3\frac{5}{9}-{\log }_3{\left(\sqrt{5}\right)}^2$

$={\log }_3\frac{5}{9}-{\log }_{3}5$$={\log }_3\left(\frac{5}{9}:5\right)$$={\log }_3{3}^{-2}=-2$

d) $4{\log }_{12}2+2{\log }_{12}3={\log }_{12}{2}^4+{\log }_{12}{3}^2$

 $={\log }_{12}\left(2^4{.3}^3\right)={\log }_{12}{\left(4.3\right)}^2$$={\log }_{12}{12}^2=2$

e) $2{\log }_{5}2-{\log }_{5}4\sqrt{10}+{\log }_5\sqrt{2}$

$={\log }_5{2}^2-{\log }_{5}4\sqrt{10}+{\log }_5\sqrt{2}$= ${\log }_{5}4-{\log }_{5}4\sqrt{10}+{\log }_5\sqrt{2}$

$={\log }_5\frac{4\sqrt{2}}{4\sqrt{10}}={\log }_5\frac{1}{\sqrt{5}}$$={\log }_5{5}^{-\frac{1}{2}}=-\frac{1}{2}$;

g) ${\log }_3\sqrt{3}-{\log }_3\sqrt[3]{9}+2{\log }_3\sqrt[4]{27}$

$={\log }_3{3}^{\frac{1}{2}}-{\log }_3{3}^{\frac{2}{3}}+2{\log }_3{3}^{\frac{3}{4}}$
$=\frac{1}{2}-\frac{2}{3}+2.\frac{3}{4}=\frac{4}{3}$