Question

Match the following expressions with there correct values.

Answer 1

Start solving each expressions one by one

$a){\log }_{0.11}0.0121$
${\log }_{0.11}0.0121={\log }_{0.11}{0.11}^2$
$\Rightarrow {\log }_{0.11}0.0121=2\left({\log }_{0.11}0.11\right)[\because \log a^m=m\log a]$
$\Rightarrow {\log }_{0.11}0.0121=2\left(1\right)[\because {\log }_{a}a=1]$
$\Rightarrow {\log }_{0.11}0.0121=2$

$b){\log }_{3\sqrt{2}}5832$
${\log }_{3\sqrt{2}}5832={\log }_{3\sqrt{2}}\left(3^6{.2}^3\right)$
$\Rightarrow {\log }_{3\sqrt{2}}5832={\log }_{3\sqrt{2}}(3\sqrt{2}{)}^6$
$\Rightarrow {\log }_{3\sqrt{2}}5832=6\left({\log }_{3\sqrt{2}}\right(3\sqrt{2}\left)\right)[\because \log a^m=m\log a]$
$\Rightarrow {\log }_{3\sqrt{2}}5832=6\left(1\right)[\because {\log }_{a}a=1]$
$\Rightarrow {\log }_{3\sqrt{2}}5832=6$

$c)2^{{\log }_{3}5}-5^{{\log }_{3}2}$
$=2^{{\log }_{3}5}-2^{{\log }_{3}5}[\because a^{{\log }_{c}b}=b^{{\log }_{c}a}]$

$\Rightarrow 2^{{\log }_{3}5}-5^{{\log }_{3}2}=0$