Question

evaluate the following​

Answer 1

$\huge\bold{\color{aqua}{Solution :-}}$

$\quad\bold{log\:\frac{75}{16}-2\:log\:\frac{5}{9}+log\:\frac{32}{243}}$

$=\bold{log\:\frac{75}{16}+log\:\frac{32}{243}-log\:(\frac{5}{9})²\qquad\therefore\tiny\orange{x\:log\:a=log\:a^x}}$

$=\bold{log\:(\frac{75}{16}×\frac{32}{243})-log\:\frac{5²}{9²}\qquad\quad\:\:\:\therefore\tiny\orange{\:log\:a+log\:b=log\:(ab)}}$

$=\bold{log\:(\frac{\red{\cancel{\blue{75}}}^{\green{\:25}}}{\red{\cancel{\purple{16}}}}×\frac{\red{\cancel{\purple{32}}}^{\green{\:2}}}{\red{\cancel{\blue{243}}}^{\green{\:81}}})-log\:\frac{25}{81}}$

$=\bold{log\:(\frac{25}{1}×\frac{2}{81})-log\:\frac{25}{81}}$

$=\bold{log\:\frac{50}{81}-log\:\frac{25}{81}}$

$=\bold{log\:\frac{\frac{50}{81}}{\frac{25}{81}}\qquad\qquad\quad\:\:\qquad\qquad\therefore\tiny\orange{\:log\:a-log\:b=log\:\frac{a}{b}}}$

$=\bold{log\:\frac{50×81}{81×25}}$

$=\bold{log\:\frac{50×\red{\cancel{\blue{81}}}}{\red{\cancel{\blue{81}}}×25}}$

$=\bold{log\:\frac{50}{25}}$

$=\bold{log\:\frac{\red{\cancel{\blue{50}}}^{\green{\:2}}}{\red{\cancel{\blue{25}}}}}$

$=\green{\boxed{\pink{\bold{log\:2}}}}$