What is the answer to x^{log_{3}(x)}=9x^{-1} ?

The answer to x^{log_{3}(x)}=9x^{-1} is x=3,x= 1/9

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x^{log_{3}(x)}=9x^{-1}

x^{l o g_{3} (x)} = 9 x^{- 1}

x = 3, x = \frac{1}{9}

Decimal Notation

x = 3, x = 0.11111 \dots

Solution steps

x^{l o g_{3} (x)} = 9 x^{- 1}

Apply log rules

lo g_{3} (x) lo g_{3} (x) = 2 - lo g_{3} (x)

Rewrite the equation with

lo g_{3} (x) = u

uu = 2 - u

Solve

uu = 2 - u : u = 1, u = - 2

u = 1, u = - 2

Substitute back

u = lo g_{3} (x),

solve for

x

Solve

lo g_{3} (x) = 1 : x = 3

Solve

lo g_{3} (x) = - 2 : x = \frac{1}{9}

x = 3, x = \frac{1}{9}

Verify Solutions:

x = 3

True

, x = \frac{1}{9}

True

The solutions are

x = 3, x = \frac{1}{9}

4 ln (x) = - 2

ln (x^{2} + 2) - ln (x - 1) = 2

15 = 6 lo g_{4} (8 x)

ln (x) - \frac{3}{ln ( x )} = 2

lo g_{3} (x + 17) - 2 = lo g_{3} (2 x)

What is the answer to x^{log_{3}(x)}=9x^{-1} ?
The answer to x^{log_{3}(x)}=9x^{-1} is x=3,x= 1/9
Solve for x: x^{log_{3}(x)}=9x^{-1}
The solution is x=3,x= 1/9