Let us determine the regular super logarithm \( \text{rslog}_b \) of \( b^x \), \( 1<b<\eta \) at the lower fixed point \( a \). Regular super logarithm shall mean that it satisfies

(1) \( \text{rslog}_b(1)=0 \)

(2) \( \text{rslog}_b(b^x)=\text{rslog}_b(x)+1 \)

and that

(3) \( \text{rslog}_b^{-1}(\text{rslog}_b(x)+t)=\exp_b^{\circ t}(x) \) where the right side is the regular iteration of \( \exp_b \) at the fixed point \( a \).

Then the formula for the principal Abel function is:

\( \alpha_b(x)=\lim_{n\to\infty} \log_{\ln(a)}(a-\exp_b^{\circ n}(x))-n \)

and that for the regular super logarithm:

\( \text{rslog}_b(x)= \alpha_b(x) - \alpha_b(1) \)

Graph of \( \text{rslog}_{\sqrt{2}} \):

Proof:

For doing this we first compute the regular Schroeder function (note that the Schroeder function is determined up to a multiplicative constant and the Abel function is determined up to an additive constant). A Schroeder function \( \sigma \) of a function \( f \) is a function that satisfies the Schroeder equation

\( \sigma(f(x))=s\sigma(x) \)

We see that we can derive a solution \( \alpha \) of the Abel equation

\( \alpha(f(x))=\alpha(x)+1 \)

by setting \( \alpha(x)=\log_s(\sigma(x)) \).

Now there is the the so called principal Schroeder function \( \sigma_f \) of a function \( f \) with fixed point 0 with slope \( s:=f'(0) \), \( 0<s<1 \) given by:

\( \sigma_f(x) = \lim_{n\to\infty} \frac{f^{\circ n}(x)}{s^n} \)

This function particularly yields the regular iteration at 0, via \( f^{\circ t}(x)=\sigma^{-1}(s^t\sigma(x)) \).

To determine the Schroeder equation at the lower fixed point \( a \) of \( \exp_b \) we consider

\( f(x)=a-b^{a-x} \) with fixed point 0 and same slope \( s=\exp_b'(a)=\ln(b)exp_b(a)=\ln(b)\log_b(a)=\ln(a)<1 \). Let \( \rho(x)=a-x \) then \( f=\rho\circ\exp_b\circ\rho=\rho^{-1}\circ\exp_b\circ\rho \).

\( f^{\circ t}(x)=\sigma_f^{-1}(s^t\sigma_f(x)) \).

\( \exp_b^{\circ t}(x)=(\rho\circ f \circ \rho^{-1})^{\circ t}=\rho\circ f^{\circ t}\circ \rho^{-1}=\rho\circ \sigma_f^{-1}\circ \mu_{s^t}\circ \sigma_f\circ \rho^{-1} \).

Hence \( \sigma_f\circ\rho^{-1} \) is the principial Schroeder function of \( \exp_b \) at \( a \).

To get the principal Abel function we take the logarithm to base \( s \):

\( \sigma_f\circ\rho^{-1}(x)=\sigma_f(a-x)=\lim_{n\to\infty} \frac{f^{\circ t}(a-x)}{s^n})=\lim_{n\to\infty} \frac{a-\exp_b^{\circ n}(a-(a-x))}{s^n}=\lim_{n\to\infty} \frac{a-\exp_b^{\circ n}(x)}{s^n} \)

\( \alpha_b(x)=\log_s(\sigma_f\circ\rho^{-1}(x))=\lim_{n\to\infty}\log_s(a-\exp_b(x))-n \).

(1) \( \text{rslog}_b(1)=0 \)

(2) \( \text{rslog}_b(b^x)=\text{rslog}_b(x)+1 \)

and that

(3) \( \text{rslog}_b^{-1}(\text{rslog}_b(x)+t)=\exp_b^{\circ t}(x) \) where the right side is the regular iteration of \( \exp_b \) at the fixed point \( a \).

Then the formula for the principal Abel function is:

\( \alpha_b(x)=\lim_{n\to\infty} \log_{\ln(a)}(a-\exp_b^{\circ n}(x))-n \)

and that for the regular super logarithm:

\( \text{rslog}_b(x)= \alpha_b(x) - \alpha_b(1) \)

Graph of \( \text{rslog}_{\sqrt{2}} \):

Proof:

For doing this we first compute the regular Schroeder function (note that the Schroeder function is determined up to a multiplicative constant and the Abel function is determined up to an additive constant). A Schroeder function \( \sigma \) of a function \( f \) is a function that satisfies the Schroeder equation

\( \sigma(f(x))=s\sigma(x) \)

We see that we can derive a solution \( \alpha \) of the Abel equation

\( \alpha(f(x))=\alpha(x)+1 \)

by setting \( \alpha(x)=\log_s(\sigma(x)) \).

Now there is the the so called principal Schroeder function \( \sigma_f \) of a function \( f \) with fixed point 0 with slope \( s:=f'(0) \), \( 0<s<1 \) given by:

\( \sigma_f(x) = \lim_{n\to\infty} \frac{f^{\circ n}(x)}{s^n} \)

This function particularly yields the regular iteration at 0, via \( f^{\circ t}(x)=\sigma^{-1}(s^t\sigma(x)) \).

To determine the Schroeder equation at the lower fixed point \( a \) of \( \exp_b \) we consider

\( f(x)=a-b^{a-x} \) with fixed point 0 and same slope \( s=\exp_b'(a)=\ln(b)exp_b(a)=\ln(b)\log_b(a)=\ln(a)<1 \). Let \( \rho(x)=a-x \) then \( f=\rho\circ\exp_b\circ\rho=\rho^{-1}\circ\exp_b\circ\rho \).

\( f^{\circ t}(x)=\sigma_f^{-1}(s^t\sigma_f(x)) \).

\( \exp_b^{\circ t}(x)=(\rho\circ f \circ \rho^{-1})^{\circ t}=\rho\circ f^{\circ t}\circ \rho^{-1}=\rho\circ \sigma_f^{-1}\circ \mu_{s^t}\circ \sigma_f\circ \rho^{-1} \).

Hence \( \sigma_f\circ\rho^{-1} \) is the principial Schroeder function of \( \exp_b \) at \( a \).

To get the principal Abel function we take the logarithm to base \( s \):

\( \sigma_f\circ\rho^{-1}(x)=\sigma_f(a-x)=\lim_{n\to\infty} \frac{f^{\circ t}(a-x)}{s^n})=\lim_{n\to\infty} \frac{a-\exp_b^{\circ n}(a-(a-x))}{s^n}=\lim_{n\to\infty} \frac{a-\exp_b^{\circ n}(x)}{s^n} \)

\( \alpha_b(x)=\log_s(\sigma_f\circ\rho^{-1}(x))=\lim_{n\to\infty}\log_s(a-\exp_b(x))-n \).