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Given $(\log_3 x)^3 = 9 \log x$, solve for $x$.

Here is what I have so far: $$(\log_3 x)^3 = \frac{9\log_3 x}{\log_3 10}$$ $$let a = \log_3 x$$ $$a^3=\frac{9a}{\log_3 10}$$ $$a^3-\frac{9a}{\log_3 10} = 0$$ $$a(a^2-\frac{9}{log_3 10}$$ $$\log_3 x = 0, \log_3 x = \pm\sqrt{\frac{9}{\log_3 10}}$$

I solved the first part of that to give $x=1$, which I plugged back in and worked. But for the second part of the solution, $x$ could equal roughly $9.743156891$ or $0.1026361385$. Plugging them both into the original equation, I get the same on both sides. Yet, when I graphed it, the only solution, as far as I could see, is $1$.

log graph

I guess my real question is, are $9.74$ and $0.10$ actual solutions to the equation? Or are the extraneous for some reason?

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You did not graph enough of it. Notice in particular the cutoffs of about $0.58$ and $1.42$ on the $x$-axis. – Jonas Meyer May 22 '12 at 3:53
up vote 2 down vote accepted

All the three are solutions. The $x$ axis in your plot covers only a limited range.

Below is the plot of $$y = \left(\log_3(x) \right)^3 - 9 \log(x)$$enter image description here

The plot was made using the software grapher on mac osx.

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Oh... I was only looking for x-axis intersections from two equations for some reason. Weirdly enough, Wolfram Alpha doesn't give me the other solutions. Thanks! – DMan May 22 '12 at 4:20

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