Why Am I Getting Incorrect Solutions to This Logarithmic Substitution Problem?

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The discussion centers on a logarithmic substitution problem where the user is confused about obtaining incorrect solutions. The key issue identified is the misunderstanding of logarithmic properties, specifically that (\log x)^3 does not equal \log (x^3). A suggested approach is to make a substitution, letting y = log x, and then solving for y before x. This method leads to a cubic equation, which should yield three solutions. The anticipated solutions are x_1=1, x_2=y, and x_3=1/y, contingent on the logarithm's base.
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(\log x)^3 \neq \log (x^3) ie. 2nd step is wrong

hint: a make substitution: y = log x and solve for y first then x.
 
The first thing you've done is to cube both sides. That's ok but it should give you
( log(x) )^3 = log(x)
Since the whole log(x) is cubed, you can't move the 3 down (that's only if the x was cubed).

But what you can do is take all the terms over to one side and then you just have to solve a cubic (which will give you 3 solutions). You may want to make it easier to see by introducing a new variable, u = log(x) for example.
 
Good to go!
 
and the solutiions should probbably be, couse i just glanced at it,:
x_1=1
x_2=y,(if the base of the logarithm is y, couse i could not see it clear)
x_3=1/y
 
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