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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$

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    Try $a = 3\sqrt[3]{a}$ and cube both sides.2018-07-28

4 Answers 4

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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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    seems a similar answer was already posted!2012-09-26
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1) Remember that $\,\log x^n=n\log x\,$

2) Now just note that $\,\log \sqrt[3] x=\log x^{1/3}\,$

Anyways, it is not true in general that $\,\log \sqrt[3] x=\sqrt[3]{\log x}\,$

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You can write $\log \sqrt[3] x$ in terms of $\log x$ using the laws of logarithms. So substitute $u=\log x$ and solve for $u$ (after rearranging you have a cubic equation in $u$) then substitute back and solve for $x$.


Edit: Your error in your working is thinking that $a^3 = \log x^3$. In fact, $a^3 = (\log x)^3$.

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    @vincentbelkin: In that case my guess is that you did some dividing, rather than factorising. You should get to $u^3=27u$, right? There are $3$ possible values $u$ can take. To find them all, factorise the expression.2012-09-26
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$\log$ is not so friendly with square/cube roots.., i.e. your identity is not true in general. Are you looking for its solution $x$?

Do you know that $\sqrt[3]x=x^{\frac13}$?

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    Posted a$n$ answer man! not sure if its right though2012-09-26