Double solutions and plotting transcendental equations I have the following transcendental equation: 
$y^2 - \log(y)^2 = 4\cdot\log(x) + 4/x + C$
and I aim to plot the equation in the positive, real quadrant, with $x>0$ (actually in the $0 < x \leq5$ interval) and $y>0$, for various values of $C$.
My computational approach was to calculate a numeric value for the R.H.S., using N number of increments in $x$'s interval and a pre-determined $C$, and then, for each of the values, solve for the corresponding value of y (using the Newton method, for e.g.) 
However, in doing this (part-successfully), it would seem that I am neglecting a second value, a sort-of double root, as it were. For e.g.:
Say $C=0$ and $x=1$, giving:
$y^2 - \log(y)^2 = 4$
I need to find two (or more?) positive solutions for y, and I'm really struggling. There's some serious self-doubt going on here and I'm just not seeing that second solution.
Big thanks in advance. 

Why not just plot it as an implicit function?  Maple, for example, has the implicitplot command.

plots[implicitplot](y^2 - log(y)^2 = 4*log(x) + 4/x + 1, x = 0 .. 5, y = 0 .. 5);


But as for your search for a second solution: $y^2 - \log(y)^2$ is increasing on $(0,\infty)$, so there is never more than one $y$ for a given $x$.
