# Iterating Log infinitely

How is it possible to find the limiting value of Log[x Log[x Log[x Log[...]]]] in Mathematica? Apparently, it should give -ProductLog[-1,-1/x], but I can't seem to replicate that ...

• If u==Log[x Log[x ...]] then u==Log[x u] and FullSimplify[ Reduce[u==Log[x u],u]] will show you the range of possibilities depending on all the values that x might have.
– Bill
Dec 2 '14 at 22:40
• Related: (66085) Dec 2 '14 at 22:48
• Letting y=Log[x*Log[x*...]] gives Exp[y]=x*Log[x*...]=x*y. Now solve for y. Dec 2 '14 at 23:00

You can use FixedPoint:

FixedPoint[Log[3.5 #] &, 3.5] == -ProductLog[-1, -1/3.5]

True


In general, if $y = f(f(...f(f(...))))$, then $y = f(y)$. Solving for $y$ will give us the formula for the infinitely nested expression.

In your case, f == Log[x #]&, which gives

sol = Refine[Reduce[y == Log[x y], y], x > 0]


Unfortunately FullSimplify can't prove Im[ProductLog[-1, -(1/x)]] >= -π for x > 0. We can extract the possible solutions for y manually:

Or @@ Cases[sol, y == f_ :> y == FullSimplify[f], Infinity]

y == -ProductLog[-1, -1/x] || y == -ProductLog[-1/x] || y == -ProductLog[1, -1/x]

• great solution - thank you! :) Dec 3 '14 at 9:20