1^infinity

How is 0*ice cream not 0? You have zero ice creams, therefore you have zero. Ice cream is perfectly finite, so it doesn't break any rules.

And in regards to the original question... yeah, it's more or less resolved.

But...

Let x = infinity

1^infinity = 1^(x), take the ln of 1^(x) (mathematically legal as long as you raise the answer to e in the end), you get

ln(1)^(x) = xln(1) = ln(1) / (1/x)

take the limit

limit x ->infinity : ln(1) / (1/x)

ln1 = 0
1/x = 0

That only proves than ln(1^infinity) is undefined.