What is the answer to x^{log_{3}(x)}=27x^2 ?

The answer to x^{log_{3}(x)}=27x^2 is x=27,x= 1/3

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x^{log_{3}(x)}=27x^2

x^{l o g_{3} (x)} = 27 x^{2}

x = 27, x = \frac{1}{3}

Decimal Notation

x = 27, x = 0.33333 \dots

Solution steps

x^{l o g_{3} (x)} = 27 x^{2}

Apply log rules

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

Rewrite the equation with

lo g_{3} (x) = u

uu = 2 u + 3

Solve

uu = 2 u + 3 : u = 3, u = - 1

u = 3, u = - 1

Substitute back

u = lo g_{3} (x),

solve for

x

Solve

lo g_{3} (x) = 3 : x = 27

Solve

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

x = 27, x = \frac{1}{3}

Verify Solutions:

x = 27

True

, x = \frac{1}{3}

True

The solutions are

x = 27, x = \frac{1}{3}

lo g_{8} (2) + lo g_{8} (4 x^{2}) = 1

lo g_{8} (x) = 3

lo g_{4} (x) + lo g_{x} (4) = 2.5

lo g_{10} (x) + lo g_{10} (8) = 2

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

What is the answer to x^{log_{3}(x)}=27x^2 ?
The answer to x^{log_{3}(x)}=27x^2 is x=27,x= 1/3
Solve for x: x^{log_{3}(x)}=27x^2
The solution is x=27,x= 1/3