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galinaleonova1
@galinaleonova1
July 2022
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Найдите наибольшее целое решение неравенства 2log[5,x]-log[x,125]<=1 (0;1/5)U(1;5 3/2)
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yarovoe
2log₅x-logₓ125≤
1
log
ₓ125=1/log₁₂₅x=1/log₅³x=3/log₅x
Пусть log₅x=у.Тогда получаем:2у-3/у-1≤0, 2у²-3-у≤0,2у²-у-3≤0,
D=1²-4·2·(-3)=25,√D=5,y₁=(1+5)/4=3/2,y₂=(1-5)/4=-1
(y+1)(y-1.5)≤0
+ - +
--------------- -1-------------------------1.5------------------------->y
-1 <log₅x<1.5
5⁻¹<x<5³/²
0.2<x<√125, 0,2<x<5√5
Ответ:11
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Answers & Comments
logₓ125=1/log₁₂₅x=1/log₅³x=3/log₅x
Пусть log₅x=у.Тогда получаем:2у-3/у-1≤0, 2у²-3-у≤0,2у²-у-3≤0,
D=1²-4·2·(-3)=25,√D=5,y₁=(1+5)/4=3/2,y₂=(1-5)/4=-1
(y+1)(y-1.5)≤0
+ - +
--------------- -1-------------------------1.5------------------------->y
-1 <log₅x<1.5
5⁻¹<x<5³/²
0.2<x<√125, 0,2<x<5√5
Ответ:11