TITLE: How do I cube/square a logarithm?
QUESTION [5 upvotes]: Btw, please don't give me the answer. I just wanna know how to raise a logarithm to its cube cause I'm stuck in this part, but don't solve it for me.
$$\log \sqrt[3]x = \sqrt[3]{\log x}$$
I tried different ways but. When I input the values in my calculator, it just doesn't match.
My answer: is it right?
  $$1/3\log x =\sqrt[3]{\log x}$$
  $$\log x = 3\sqrt[3]{\log x}$$
  $$a = 3\sqrt[3]{\log x}$$
  $$a^3 = \log x^{27} $$
  $$\log x^3 = \log x^{27}$$
  $$\log x^{24} = 0$$
  $$ x = \sqrt[24]{1}$$
  $$ x = 1$$

REPLY [0 votes]: One method you may attempt; when you get to the step:
$\log(x)^3=27\log(x)$
Let $u= \log(x)$
and solve cubic:  $ u^3 - 27u = 0$
so that: $u(u^2 - 27)$=0
therefore: $u(u-\sqrt{27})(u + \sqrt{27})= 0$
Now find the roots then resubstitute $\log(x) = u$; to get the answer
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