Помогите. очень срочно!!
1)Log с основанием 1/2. 1/42.
2)Log с основанием 1/2. (tg(п/7))+Log с основанием 1/2. (tg(5/14)п).
3)Log с основанием sqrt2. (sin(п/8)+Log с основанием sqrt2. (2cos(п/8).
Ответы
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1) ![log_{ frac{1}{2}} frac{1}{42}=log_{2^{-1}}42^{-1}=log_242=log_2(2*21)=1+log_221; log_{ frac{1}{2}} frac{1}{42}=log_{2^{-1}}42^{-1}=log_242=log_2(2*21)=1+log_221;](https://tex.z-dn.net/?f=log_%7B+frac%7B1%7D%7B2%7D%7D+frac%7B1%7D%7B42%7D%3Dlog_%7B2%5E%7B-1%7D%7D42%5E%7B-1%7D%3Dlog_242%3Dlog_2%282%2A21%29%3D1%2Blog_221%3B+)
2)![log_{ frac{1}{2}}(tg( frac{ pi }{7}))+log_{ frac{1}{2}}(tg( frac{5 pi }{14}))= \ \ log_{ frac{1}{2}}( frac{cos( frac{ pi }{7}- frac{5 pi }{14})-cos(frac{ pi }{7}+frac{5 pi }{14})}{cos( frac{ pi }{7}- frac{5 pi }{14})+cos(frac{ pi }{7}+frac{5 pi }{14})} )=log_{ frac{1}{2}} ( frac{cos(- frac{3 pi }{14})-cos frac{ pi }{2}}{cos(- frac{3 pi }{14})+cos frac{ pi }{2}})=log_{ frac{1}{2}}( frac{cos( frac{3 pi }{14}-0 )}{cos( frac{3 pi }{14}+0 })= log_{ frac{1}{2}}(tg( frac{ pi }{7}))+log_{ frac{1}{2}}(tg( frac{5 pi }{14}))= \ \ log_{ frac{1}{2}}( frac{cos( frac{ pi }{7}- frac{5 pi }{14})-cos(frac{ pi }{7}+frac{5 pi }{14})}{cos( frac{ pi }{7}- frac{5 pi }{14})+cos(frac{ pi }{7}+frac{5 pi }{14})} )=log_{ frac{1}{2}} ( frac{cos(- frac{3 pi }{14})-cos frac{ pi }{2}}{cos(- frac{3 pi }{14})+cos frac{ pi }{2}})=log_{ frac{1}{2}}( frac{cos( frac{3 pi }{14}-0 )}{cos( frac{3 pi }{14}+0 })=](https://tex.z-dn.net/?f=log_%7B+frac%7B1%7D%7B2%7D%7D%28tg%28+frac%7B+pi+%7D%7B7%7D%29%29%2Blog_%7B+frac%7B1%7D%7B2%7D%7D%28tg%28+frac%7B5+pi+%7D%7B14%7D%29%29%3D+%5C++%5C+log_%7B+frac%7B1%7D%7B2%7D%7D%28+frac%7Bcos%28+frac%7B+pi+%7D%7B7%7D-+frac%7B5+pi+%7D%7B14%7D%29-cos%28frac%7B+pi+%7D%7B7%7D%2Bfrac%7B5+pi+%7D%7B14%7D%29%7D%7Bcos%28+frac%7B+pi+%7D%7B7%7D-+frac%7B5+pi+%7D%7B14%7D%29%2Bcos%28frac%7B+pi+%7D%7B7%7D%2Bfrac%7B5+pi+%7D%7B14%7D%29%7D+%29%3Dlog_%7B+frac%7B1%7D%7B2%7D%7D+%28+frac%7Bcos%28-+frac%7B3+pi+%7D%7B14%7D%29-cos+frac%7B+pi+%7D%7B2%7D%7D%7Bcos%28-+frac%7B3+pi+%7D%7B14%7D%29%2Bcos+frac%7B+pi+%7D%7B2%7D%7D%29%3Dlog_%7B+frac%7B1%7D%7B2%7D%7D%28+frac%7Bcos%28+frac%7B3+pi+%7D%7B14%7D-0+%29%7D%7Bcos%28+frac%7B3+pi+%7D%7B14%7D%2B0+%7D%29%3D)
![log_{ frac{1}{2}}1=0 log_{ frac{1}{2}}1=0](https://tex.z-dn.net/?f=+log_%7B+frac%7B1%7D%7B2%7D%7D1%3D0)
3)![log_{ sqrt{2}}(sin frac{ pi }{8})+log_{ sqrt{2}}(2cos frac{ pi }{8})=log_{ sqrt{2}}(2sin frac{ pi }{8}cos frac{ pi }{8})=log_{ sqrt{2}}sin frac{ pi }{4}= \ log_{ sqrt{2}}( frac{ sqrt{2}}{2})=log_{ sqrt{2}} sqrt{2}-log_{ sqrt{2} }2=1-log_{ sqrt{2}}( sqrt{2})^2=1-2log_{ sqrt{2}} sqrt{2}= log_{ sqrt{2}}(sin frac{ pi }{8})+log_{ sqrt{2}}(2cos frac{ pi }{8})=log_{ sqrt{2}}(2sin frac{ pi }{8}cos frac{ pi }{8})=log_{ sqrt{2}}sin frac{ pi }{4}= \ log_{ sqrt{2}}( frac{ sqrt{2}}{2})=log_{ sqrt{2}} sqrt{2}-log_{ sqrt{2} }2=1-log_{ sqrt{2}}( sqrt{2})^2=1-2log_{ sqrt{2}} sqrt{2}=](https://tex.z-dn.net/?f=log_%7B+sqrt%7B2%7D%7D%28sin+frac%7B+pi+%7D%7B8%7D%29%2Blog_%7B+sqrt%7B2%7D%7D%282cos+frac%7B+pi+%7D%7B8%7D%29%3Dlog_%7B+sqrt%7B2%7D%7D%282sin+frac%7B+pi+%7D%7B8%7Dcos+frac%7B+pi+%7D%7B8%7D%29%3Dlog_%7B+sqrt%7B2%7D%7Dsin+frac%7B+pi+%7D%7B4%7D%3D+%5C+log_%7B+sqrt%7B2%7D%7D%28+frac%7B+sqrt%7B2%7D%7D%7B2%7D%29%3Dlog_%7B+sqrt%7B2%7D%7D+sqrt%7B2%7D-log_%7B+sqrt%7B2%7D+%7D2%3D1-log_%7B+sqrt%7B2%7D%7D%28+sqrt%7B2%7D%29%5E2%3D1-2log_%7B+sqrt%7B2%7D%7D+sqrt%7B2%7D%3D+++++++)
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