how to find the number of solutions for $\ln(x) =x^2/8$ using simple math principles? how to find the number of solutions for  $\ln(x) = x^2/8$ with high school level math (basic logarithm rules or something of that nature)( and without graphing $\ln(x)$ and $x^2/8$). (sorry for the poor formatting, feel free to edit)
I tried to change the equation to someting familiar to use a logarithm rule:
$$\displaylines{
x^2/8 = \ln(x) \cr
x/8^{0.5} = lan(x)^{0.5} \cr
x = 8^{0.5}*lan(x)^{0.5}
}$$
Changing the representation of the equation using the basic logarithm rule
$$\displaylines{
x^2/8 = \ln(x) \cr
x = e^{x^2/8}
}$$
But my tries got me nowhere. Any suggestions?
Let $y:=x^2$ (as $x&gt;0$, no solution is introduced). The equation can be written
$$y=e^{y/4}.$$
The exponential is a convex function, so it has at most two intersections with a straight line.
Now as $0&lt;e^{0/4},4&gt;e^{4/4}$ and $12&lt;e^{12/4}$, we have these two intersections.
try visualizing the graphs of ln(x) and x^2 / 8. The first is a log-curve, and the second in a parabola. They will  intersect at at most two points. so you will have at most two solutions.
Since the log-curve starts (or ends) at the origin, and the parabola will &quot;circumnavigate&quot; the origin going from -infinity to +infinity, it's safe to say that they will intersect at exactly one point.