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ОДЗ: 9x>0 9x≠1 x≠1
x>0 x≠1/9
x∈(0; 1/9)U(1/9; 1)U(1; +∞)
![log_{9x}9-log_{9x}x+ frac{1}{log_{x}^29}=1 \ \
frac{1}{log_{9}9x}- frac{1}{log_{x}9x}+ frac{1}{log_{x}^29}=1 \ \
frac{1}{log_{9}9+log_{9}x}- frac{1}{log_{x}9+log_{x}x}+ frac{1}{log_{x}^29}=1 \ \
frac{1}{1+ frac{1}{log_{x}9} }- frac{1}{1+log_{x}9}+ frac{1}{log_{x}^29}=1](https://tex.z-dn.net/?f=log_%7B9x%7D9-log_%7B9x%7Dx%2B+frac%7B1%7D%7Blog_%7Bx%7D%5E29%7D%3D1+%5C++%5C+%0A+frac%7B1%7D%7Blog_%7B9%7D9x%7D-+frac%7B1%7D%7Blog_%7Bx%7D9x%7D%2B+frac%7B1%7D%7Blog_%7Bx%7D%5E29%7D%3D1+%5C++%5C+%0A+frac%7B1%7D%7Blog_%7B9%7D9%2Blog_%7B9%7Dx%7D-+frac%7B1%7D%7Blog_%7Bx%7D9%2Blog_%7Bx%7Dx%7D%2B+frac%7B1%7D%7Blog_%7Bx%7D%5E29%7D%3D1+%5C++%5C+%0A+frac%7B1%7D%7B1%2B+frac%7B1%7D%7Blog_%7Bx%7D9%7D+%7D-+frac%7B1%7D%7B1%2Blog_%7Bx%7D9%7D%2B+frac%7B1%7D%7Blog_%7Bx%7D%5E29%7D%3D1++++++++++ "log_{9x}9-log_{9x}x+ frac{1}{log_{x}^29}=1 \ \
frac{1}{log_{9}9x}- frac{1}{log_{x}9x}+ frac{1}{log_{x}^29}=1 \ \
frac{1}{log_{9}9+log_{9}x}- frac{1}{log_{x}9+log_{x}x}+ frac{1}{log_{x}^29}=1 \ \
frac{1}{1+ frac{1}{log_{x}9} }- frac{1}{1+log_{x}9}+ frac{1}{log_{x}^29}=1")
![y=log_{x}9 \ \
frac{1}{1+ frac{1}{y} }- frac{1}{1+y}+ frac{1}{y^2}=1 \ \
frac{y}{1+y}- frac{1}{1+y}+ frac{1}{y^2}=1 \ \
y^3-y^2+1+y=y^2(1+y) \
y^3-y^3-y^2-y^2+y+1=0 \
-2y^2+y+1=0 \
2y^2-y-1=0 \
D=1+8=9 \
y_{1}= frac{1-3}{4} = -frac{2}{4}=- frac{1}{2} \
y_{2}= frac{1+3}{4}=1](https://tex.z-dn.net/?f=y%3Dlog_%7Bx%7D9+%5C++%5C+%0A+frac%7B1%7D%7B1%2B+frac%7B1%7D%7By%7D+%7D-+frac%7B1%7D%7B1%2By%7D%2B+frac%7B1%7D%7By%5E2%7D%3D1+%5C++%5C+%0A+frac%7By%7D%7B1%2By%7D-+frac%7B1%7D%7B1%2By%7D%2B+frac%7B1%7D%7By%5E2%7D%3D1+%5C++%5C+%0Ay%5E3-y%5E2%2B1%2By%3Dy%5E2%281%2By%29+%5C+%0Ay%5E3-y%5E3-y%5E2-y%5E2%2By%2B1%3D0+%5C+%0A-2y%5E2%2By%2B1%3D0+%5C+%0A2y%5E2-y-1%3D0+%5C+%0AD%3D1%2B8%3D9+%5C+%0Ay_%7B1%7D%3D+frac%7B1-3%7D%7B4%7D+%3D+-frac%7B2%7D%7B4%7D%3D-+frac%7B1%7D%7B2%7D+%5C+%0Ay_%7B2%7D%3D++frac%7B1%2B3%7D%7B4%7D%3D1++++++++ "y=log_{x}9 \ \
frac{1}{1+ frac{1}{y} }- frac{1}{1+y}+ frac{1}{y^2}=1 \ \
frac{y}{1+y}- frac{1}{1+y}+ frac{1}{y^2}=1 \ \
y^3-y^2+1+y=y^2(1+y) \
y^3-y^3-y^2-y^2+y+1=0 \
-2y^2+y+1=0 \
2y^2-y-1=0 \
D=1+8=9 \
y_{1}= frac{1-3}{4} = -frac{2}{4}=- frac{1}{2} \
y_{2}= frac{1+3}{4}=1")
При y=-1/2
![log_{x}9=- frac{1}{2} \ \
x^{- frac{1}{2} }=9 \ \
frac{1}{ sqrt{x} } =9 \ \
sqrt{x} = frac{1}{9} \ \
x= frac{1}{81}](https://tex.z-dn.net/?f=log_%7Bx%7D9%3D-+frac%7B1%7D%7B2%7D+%5C++%5C+%0Ax%5E%7B-+frac%7B1%7D%7B2%7D+%7D%3D9+%5C++%5C+%0A+frac%7B1%7D%7B+sqrt%7Bx%7D+%7D++%3D9+%5C++%5C+%0A+sqrt%7Bx%7D+%3D+frac%7B1%7D%7B9%7D+%5C++%5C+%0Ax%3D+frac%7B1%7D%7B81%7D++ "log_{x}9=- frac{1}{2} \ \
x^{- frac{1}{2} }=9 \ \
frac{1}{ sqrt{x} } =9 \ \
sqrt{x} = frac{1}{9} \ \
x= frac{1}{81}")
При y=1
![log_{x}9=1 \
x=9](https://tex.z-dn.net/?f=log_%7Bx%7D9%3D1+%5C+%0Ax%3D9 "log_{x}9=1 \
x=9")
k=2
x₀=9
Ответ: 2) 10.
x>0 x≠1/9
x∈(0; 1/9)U(1/9; 1)U(1; +∞)
При y=-1/2
При y=1
k=2
x₀=9
Ответ: 2) 10.
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