What is the answer to x^{log_{3}(x)}=81x^3 ?

The answer to x^{log_{3}(x)}=81x^3 is x=81,x= 1/3

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x^{log_{3}(x)}=81x^3

x^{l o g_{3} (x)} = 81 x^{3}

x = 81, x = \frac{1}{3}

Decimal Notation

x = 81, x = 0.33333 \dots

Solution steps

x^{l o g_{3} (x)} = 81 x^{3}

Apply log rules

lo g_{3} (x) lo g_{3} (x) = 3 lo g_{3} (x) + 4

Rewrite the equation with

lo g_{3} (x) = u

uu = 3 u + 4

Solve

uu = 3 u + 4 : u = 4, u = - 1

u = 4, u = - 1

Substitute back

u = lo g_{3} (x),

solve for

x

Solve

lo g_{3} (x) = 4 : x = 81

Solve

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

x = 81, x = \frac{1}{3}

Verify Solutions:

x = 81

True

, x = \frac{1}{3}

True

The solutions are

x = 81, x = \frac{1}{3}

lo g_{2} (27) = 3 lo g_{2} (x)

(ln (x))^{2} - ln (x^{7}) = 8

- lo g_{10} (10 x) = 11.45

2 lo g_{10} (x + 86) = 4

lo g_{4} (x) - 2 = 2

What is the answer to x^{log_{3}(x)}=81x^3 ?
The answer to x^{log_{3}(x)}=81x^3 is x=81,x= 1/3
Solve for x: x^{log_{3}(x)}=81x^3
The solution is x=81,x= 1/3