I have the follwing equation:

$$y=\frac 1 4x^2 -\frac 1 2 \ln{x}$$

How can $x$ be expressed in terms of $y$?

  • $\begingroup$ This isn't possible in terms of elementary functions. $\endgroup$ – user217285 Jul 4 '15 at 5:26
  • $\begingroup$ what is your question write it correctly $\endgroup$ – absolute friend Jul 4 '15 at 5:28
  • $\begingroup$ I was not satisfied with the formulation of the problem. My English is not very good, but I think 'express' is an appropriate term . If it not, please ´help tofind the right word. $\endgroup$ – miracle173 Jul 4 '15 at 5:42
  • $\begingroup$ This is decreasing from $0$ to $1$, then increasing, In general there will be two $x$ for every $y$. And, as has already been observed, we cannot solve using elementary functions. $\endgroup$ – André Nicolas Jul 4 '15 at 5:54

We have

$$y=\frac14 x^2-\frac12 \log x\tag 1$$

Multiplying $(1)$ by $-4$ yields

$$\begin{align} -4y&=- x^2+2 \log x\\\\ &=-x^2+\log x^2 \tag 2\\\\ \end{align}$$

Now, exponentiating both sides of $(2)$ and multiplying by $-1$ reveals that

$$-e^{-4y}=-x^2e^{-x^2} \tag 3$$

from which we find that


where $W(z)$ is the Lambert W Function. Note we rejected negative values of $x$ since in $(1)$ we have $\log x$.

We must have $W\le 0$ and $-e^{-4y}\ge -e^{-1}\implies y\ge \frac14$.

  • $\begingroup$ You are too fast for the old man ! Cheers :-) $\endgroup$ – Claude Leibovici Jul 4 '15 at 6:13
  • $\begingroup$ @ClaudeLeibovici Well ... I might be close to you there ... and slow. Plus 3 days out of surgery!! $\endgroup$ – Mark Viola Jul 4 '15 at 6:15
  • $\begingroup$ I wish you a prompt recovery ! $\endgroup$ – Claude Leibovici Jul 4 '15 at 6:20
  • $\begingroup$ @ClaudeLeibovici Thank you!! Forth surgery in a year and the second in 4 months. $\endgroup$ – Mark Viola Jul 4 '15 at 6:21

As said in comments, the solution of equation $$\frac 1 4x^2 -\frac 1 2 \ln{x}-y=0$$ cannot be expressed in terms of elementary functions.

However, there are explicit solutions in terms of Lambert function. Chenging variable $x^2=t$, the equation write $$t-\log(t)-4y=0$$ and the solution is $$t=-W\left(-e^{-4 y}\right)$$

You must notice that $y'=\frac{x}{2}-\frac{1}{2 x}$ cancels at $x=1$ and, at this point $y=\frac 14$ and the second derivative is always positive. So, if $y<\frac 14$, there is no real root; if $y=\frac 14$, the only root is $x=1$ and if $y>\frac 14$, there are two real roots (one between $0$ and $1$, another larger than $1$).

  • $\begingroup$ Not so slow after all! $\endgroup$ – Mark Viola Jul 4 '15 at 6:21

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