Solving $\log_2(x^2) = x$ explicitly?

Question

I'm having problems getting a proper step-by-step solution to this equation. $$ \log_2(x^2) = x $$ I know the results are 2 and 4, but so far I can get only solutions like these: $$ 2^x = x^2 \qquad \text{or} \qquad \frac{\log(x)}{x} = \frac{\log(2)}{2}. $$

I'd like something along the lines of $x = \dotsb$.

Is it possible with some basic knowledge?

The real solutions include the product-log function according to WA. You can't solve it in terms of elementary algebra. — Shahar, Jan 13 '15 at 18:30
we get $x=2$ or $x=4$ and after this you need the Lambert function — Dr. Sonnhard Graubner, Jan 13 '15 at 18:30

score 2 · Answer 1 · answered Jan 13 '15 at 18:30

No. There is in fact a third solution, $x\approx -0.77$, but you cannot find it using elementary algebraic methods. The best you can do is to "guess" these solutions (by inspection in the case of the positive ones, or by using a numerical method) and then prove using Calculus that there can be no others (by comparing the growth rates of the two functions).

Solving $\log_2(x^2) = x$ explicitly?

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