Please....please help with indices and logarithms!!?

Question

I'm quite confusing with indices..especially with this question..
x ^ 1/3=81x

(the sign ^ means to the power of)
hopefully ,all of you can help me..please

Answer 1

x^1/3 = 81x

Change 81 into exponential 3^4
x^1/3 = 3^4 x

Raise both sides to the power of 3
x = 3^12 x³

Transpose x
3^12x³ - x = 0

Factor out
x(3^12 x² - 1) = 0

Factor out more
x(3^6 x - 1)(3^6 x + 1) = 0

The zero product property
x = 0 or 3^6 x - 1 = 0 or 3^6 x + 1 = 0

transpose
x = 0 or 3^6 x = 1 or 3^6 x = -1

Divide
x = 0 or x = 1/3^6 or x = -1/3^6

Change 3^6 = 729
x = 0 or x = 1/729 or x = -1/729

Therefore, the values for x are:
x = 0
x = 1/729
x = -1/729

Answer 2

x^1/3 = 81x..........raise to power 3
X = 81³x³
531441x³ - x = 0
x (531441x² - 1 ) = 0
so
x = 0
or
531441x² - 1 = 0
x² = 1/531441 = 531441 ^ -1.....take the square root
x = ± 531441 ^ -1/2
x = ± 729^-1
x = ± 1/729

so
x Є { 0 , 1/729 , -1/729 }

Answer 3

Or, to use logarithms;

x^1/3=81x
1/3*logx=log81+logx
-2/3*logx=log81
logx=-3/2*log81
x=10^(-3/2*log81)