Change of base formula for Tetration

[jaydfox]

Then I dont understand how you compute \( \log_a^{\circ n} AB=\log_a^{\circ n}(A)+\epsilon_1 \).

This is one of the logarithmic laws:

\( \log(AB) = \log(A)+\log(B) \)

Then, we know that \( \log(1)=0 \), and that \( \log(1+\epsilon) \approx \log_b(e)\ \times\ \epsilon \). As epsilon goes to 0, this is in fact exact.

Therefore, if B is (1+epsilon), then we can state:
\( \log(AB) = \log(A) + \log(B) = \log(A) + \epsilon \)

Never mind, I see what you meant. The epsilon needs to be inside the parentheses. I'll post an update when I get a chance.

Essentially, the correction is:

\( \log_a^{\circ n} AB=\log_a^{\circ (n-1)}\left(\log_a(A)+\epsilon_1\right) \)

Again, it doesn't affect the limit case.