Question

Логарифмы. Задание на фото.

Answer 1

$frac{log_336}{log_43} - frac{log_312}{log_{12}3}= log_3(3cdot3cdot4)log_34-log_312log_312= \ =(log_33+log_33+log_34)log_34-(log_3(4cdot3))^2= \ =(1+1+log_34)log_34-(log_34+log_33)^2= \ =(2+log_34)log_34-(log_34+1)^2= \ =2log_34+(log_34)^2-(log_34)^2-2log_34-1=-1$

$log_{ frac{1}{2} }(cos frac{ pi }{6} -sin frac{ pi }{6} )+log_{ frac{1}{2} }(cos frac{ pi }{6}+sin frac{ pi }{6} )= \ =log_{ frac{1}{2} }(cos frac{ pi }{6} -sin frac{ pi }{6} )(cos frac{ pi }{6}+sin frac{ pi }{6} )= \ =log_{ frac{1}{2} }(cos^2 frac{ pi }{6} -sin ^2frac{ pi }{6} )= log_{ frac{1}{2} }(cos frac{ pi }{3} )=log_{ frac{1}{2} }frac{1}{2}=1 \ log_{13}x=1 \ x=13^1=13$

$f(x)=x^{5+log_2x}-15^{log_{15}2^{-4}}= x^5cdot x^{log_2x}-2^{-4}= \ =x^5cdot x^{log_2x}-2^{-4}= x^5cdot x^{log_2x}-2^{-4} \ f( frac{1}{2} )=(frac{1}{2})^5cdot (frac{1}{2})^{log_2frac{1}{2}}-2^{-4}= \ =2^{-5}cdot (frac{1}{2})^{-1}-2^{-4}=2^{-5}cdot 2-2^{-4}=2^{-4}-2^{-4}=0$

$11^{log_{sqrt{11}}2}+log_3 frac{5+2sqrt{6}}{9}-log_frac{1}{sqrt{3}}( sqrt{3}-sqrt{2})= \ =sqrt{11}^{2log_{ sqrt{11}}2}+log_3 frac{5+2 sqrt{6}}{9}-log_{sqrt{3}}(frac{1}{sqrt{3} - sqrt{2}})= \ =sqrt{11}^{log_{ sqrt{11}}2^2}+log_3 frac{5+2 sqrt{6}}{9}-log_{sqrt{3}}(frac{sqrt{3}+ sqrt{2}}{(sqrt{3}-sqrt{2})(sqrt{3}+sqrt{2})})=$
$2^2+log_3 frac{5+2 sqrt{6}}{9}-log_{sqrt{3}}(sqrt{3}+ sqrt{2})= \ =4+log_3 frac{5+2 sqrt{6}}{9}-log_3(sqrt{3}+ sqrt{2})^2= \ =4+log_3 frac{5+2 sqrt{6}}{9}-log_3(3+2+2sqrt{6}})= \ =4+log_3 frac{5+2 sqrt{6}}{9(5+2sqrt{6})}=4+log_3 frac{1}{9}=4-2=2$

$(m^{ frac{log_4n}{log_2n}}cdot n^{ frac{log_4n}{log_2n}})^{2log_{mn}3}= ((mn)^{ frac{log_4n}{log_2n}}}})^{2log_{mn}3}= \ =((mn)^{ frac{log_n2}{log_n4}}}})^{2log_{mn}3}=((mn)^{log_42}}})^{2log_{mn}3}= \ =((mn)^{0.5}}}})^{2log_{mn}3}=(mn}}})^{log_{mn}3}=3$

$log_{ pi ^2} sqrt{a} =1 \ sqrt{a} = pi ^2 \ a= pi ^4 \ log_{ pi ^2} b =1 \ b= pi ^2 \ log_{ pi ^3}( frac{ab^3}{ pi } )=log_{ pi ^3}( frac{ pi ^4( pi ^2)^3}{ pi } )=log_{ pi ^3} pi ^9=3$

$3^{log_9(x+2 sqrt{x-2}-1)}+7^{log_{49}(x-2 sqrt{x-2}-1)}= \ =3^{log_9(x-2+2sqrt{x-2}+1) }+7^{log_{49}(x-2-2sqrt{x-2}+1)}= \ =3^{log_9((sqrt{x-2})^2+2 sqrt{x-2}+1)}+7^{log_{49}(( sqrt{x-2})^2 -2 sqrt{x-2}+1 )}= \ =3^{log_{3^2}( sqrt{x-2}+1)^2 }+7^{log_{7^2}(sqrt{x-2}-1)^2}= \ =3^{log_{3}| sqrt{x-2}+1|}+7^{log_{7}|sqrt{x-2}-1| }= \ = |sqrt{x-2}+1|+|sqrt{x-2}-1|= |sqrt{2.01-2}+1|+|sqrt{2.01-2}-1|= \ =|0.1+1|+|0.1-1|=|1.1|+|-0.9|=1.1+0.9=2$