Question

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Answer 1

$\displaystyle \mathtt{a)~log_{12}8+log_{12}18=log_{12}(8 \cdot 18)=log_{12}144=log_{12}12^2=2log_{12}12=2}$

$\displaystyle \mathtt{b)~log_425-2log_45=log_425-log_45^2=log_425-log_425=}\\ \\ \mathtt{=log_4 \frac{25}{25}=log_41=0 }$

$\displaystyle \mathtt{d)~log_{36}84-log_{36}14=log_{36} \frac{84}{14}=log_{36} 6=log_{6^2}6= \frac{1}{2}log_66= \frac{1}{2} }$

$\displaystyle \mathtt{e)~log_{49}84-log_{49}12=log_{49} \frac{84}{12}=log_{49} 7=log_{7^2}7= \frac{1}{2}log_77= \frac{1}{2} }$

$\displaystyle \mathtt{g)~3 lg5+ \frac{1}{2} lg64=lg5^3+lg64^{ \frac{1}{2} }=lg125+lg \sqrt{64} =}\\ \\ \mathtt{=lg125+lg8=lg(125 \cdot 8)=lg1000=lg10^3=3lg10=3}$

$\displaystyle \mathtt{h)~log_{ \sqrt{2} }12-log_29=log_{2^{ \frac{1}{2} }}12-log_29= \frac{1}{ \frac{1}{2}}log_212-log_29=}\\ \\ \mathtt{=2log_212-log_29=log_212^2-log_29=log_2144-log_29=}\\ \\ \mathtt{=log_2 \frac{144}{9} =log_216=log_22^4=4log_22=4}$

$\displaystyle \mathtt{j)~log_{ \sqrt{3} }24-log_94^6=log_{3^{ \frac{1}{2} }}24-6log_{3^2}4= \frac{1}{ \frac{1}{2} }log_324- \frac{1}{2}\cdot 6log_34= }\\ \\ \mathtt{=2log_324-3log_34=log_324^2-log_34^3=log_3576-log_364=}\\ \\ \mathtt{=log_3 \frac{576}{64}=log_3 9=log_33^2=2log_33=2}$

$\displaystyle \mathtt{k)~log_{ \sqrt{2} }54-log_49^6=log_{2^{ \frac{1}{2} }}54-6log_{2^2}9= \frac{1}{ \frac{1}{2} }log_254- \frac{1}{2}\cdot 6log_29= }\\ \\ \mathtt{=2log_254-3log_29=log_254^2-log_29^3=log_22916-log_2729=}\\ \\ \mathtt{=log_2 \frac{2916}{729}=log_2 4=log_22^2=2log_22=2}$