# Solve $3^y=y^3,\space y\neq1,\space y\neq3$

I was messing around the other day and I noticed this:
$2^3<3^2$
$2^4=4^2$
$2^5>5^2$

and I wondered if there is a pattern, i.e.
$3^x<x^3$
$3^y=y^3$
$3^z>z^3$
$x=y-1=z-2,\space y\neq1,\space y\neq3$
Then solve for $y$, and repeat with larger integers in place of the 3 until I can find a pattern.

The only way I know to solve this is with iterative equations. The problem is, I can't find an iterative equation and starting value combination that give the right answer.

Also, I have no idea what to tag this question.

• $y \approx 2.478$ Jul 1, 2014 at 10:30

Graphs are useful in such situations, if you would like to reduce your calculation efforts. I plot the graph, it looks absolutely wavy and it intersects the x axis at 2 points like below I tried zooming in and calculating the value of the first intersect it comes out $y\approx2.4781$

Let us suppose that you do not know Lambert function and that you want to solve $$f(y)=3^y-y^3=0$$

If you plot the function, you notice that there is a root between $2$ and $3$. So, use Newton method, which, starting with a "reasonable" guess $y_0$, will update it according to $$y_{n+1}=y_n-\frac{f(y_n)}{f'(y_n)}$$ So, let us start with $y_0=2$. Newton iterates will then be successively : $2.47338$, $2.47803$, $2.47805$ which is the solution for six significant figures.

Please notice that any equation of the form $$A+Bx+C\log(D+Ex)=0$$ has a closed form solution using Lambert function.

If we look at the solution of the more general equation $$f(y)=a^y-y^a=0$$ the solution is given by $$y=-\frac{a W\left(-\frac{\log (a)}{a}\right)}{\log (a)}$$ and, then, deside the trivial solution $y=a$, a second root exist if and only if $a>e$. This second solution is a decreasing function of $a$, starting at $e$ and decreasing asymptotically to $1$ if $a$ goes to infinity.
• You missed out on a $and I can't submit edit! :( – MonK Jul 1, 2014 at 11:12 • @Sid. Thank you ! Cheers :) Jul 1, 2014 at 11:21$y=-3\frac{W(-\frac{\ln(3)}{3})}{\ln(3)}$The Lambert W function$W(x)$is multivaluated in the range$-\frac{1}{e}<x<0x=-\frac{\ln(3)}{3}$The two roots are$y=3$and$y=2.47805...\$