What is x when x^(x^3)=3?

Question

Solve for x.
X to the power of (X to the power of 3) = 3

I know the answer is cubed root of 3, I need to know HOW you did it. You use logarithms, right? And then what? I'm so close.

Answer 1

Take the Log of both sides:
Log(x^3)=Log(3)
Log(x^3) = 3Log(x)= Log(3).
Log(x) = 1/3*Log(3) = Log(3^(1/3))
Take the exp of both sides
exp(Log(x)) = exp [Log(3^(1/3))]
This implies
x = 3^(1/3)

Answer 2

x^(x^3)=3

It is interesting that you know THE answer is cubed root of 3
and that there might not be other answers also. After all a cubic equation has 3 roots. How many roots does a 'x-ic' equation have.?

The HOW of solution involves the LambertW function.
See http://mathworld.wolfram.com/LambertW-Function.html

Sketchily▬
x^(x^3)=(x^x)^3=3
x^x=3^(1/3)=k
x*ln(x)=ln(k)
x= ln(k)/ln(x)
The only way to solve this equation is through successive approximations which brings us to the LambertW function
which is something like a super-log table, difficult to construct,
or find, or if done. Here,I am out of gas.

In your problem, x=3^(1/3) was a sheer guess that worked.
What if x^(x^3)= 4? Any guesses?

Answer 3

x^(x^3) = 3
logbase3 x^(x^3) = logbase3 3
x^3 * logbase3 x = logbase3 3
x^3 * logbase3 x = 1
logbase3 x = x^-3
x = 3^x^-3
x = 3/(x^-3)
cuberootx = cuberoot3 / cuberoot x^-3
cuberoot(x*x^-3) = cuberoot3
cuberoot(x^-2) = cuberoot3
x = cuberoot 3

Good luck!

Answer 4

3 = [3^(1/3)]^3 = [3^(1/3)]^[3^(1/3)]^3
x = 3^(1/3)