# Tetration Forum

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I've been sort of obsessed with Tetratipn since I first heard about it, and I've scoured the internet trying to learn all that I could about it. I came across this oeis Talk: Iterated Logarithms recently and the idea of a weighted function to deal with non-Natural numbers struck me as interesting. I've played with his idea and came up with this:
$^ya = {a_{_0}}^.^{.}^.^{(a_{h-1} - ((a - 1) - r))}$
and $log_b*(^ya) = \lceil slog_b(^ya) \rceil = \lceil y \rceil$

$slog_b(^ya) = log_b*(^ya) - r$

Where $r = y - \lfloor y \rfloor$ and $h = \lfloor y + 1 \rfloor$

This leads to a continuous function that looks like a very steep exponential graph.

It also leads to several identities:
$slog_b(^ya) = yslog_b(a)$

$^0a = a_0 - (a - 1) = a - a + 1 = 1;$$h = \lfloor 0 + 1 \rfloor = 1$

$^{-y}a = \frac {1}{a_{_0}^.^{.}^.^{(a_{h-1} - ((a - 1) - r))}}$