Hey, it's a change of base formula for **tetration**. Some people only define tetration for integers \( n \ge 1 \), others for integers \( n \ge 0 \), others for integers \( n \ge -1 \). Some people define it for real numbers \( x \ge 0 \), or \( x \ge -1 \) or \( x > -2 \). Some people might define tetration over all reals, or over the complex numbers. I prefer reals greater than -2 myself, but your mileage may vary.
For real numbers, my formula is valid, assuming you define tetration as iterated exponentials, with negative iterations equivalent to logarithms. I don't claim validity over complex numbers, mainly because the formula's very basis for convergence is the increasing modulus of the successive tetrations, and complex tetrations can cause the modulus to decrease, which invalidates the limit.
For real x, use whatever domain you use for tetration. If *you* define tetration as valid for x>-2, what did you expect to happen when you put in x <= -2? The problem isn't with my formula, it's with your indecision on what the valid domain for x is. Use whatever domain you consider valid.
Moving along:
There are two main base conversion formulae for exponentiation for base a, given an exact solution for base b:
\(
\begin{eqnarray}
a^x & = & b^{\log_b(a^x)} \\
\\[5pt]
\\
a^x & = & b^{\log_b(a)\times x} \\
\end{eqnarray}
\)
The first is a trivial restatement of the definition of log_b(z) as the inverse function of b^z. The second displays some "fundamental truth" about exponentiation that isn't obvious from looking solely at the first formula. It allows you to solve for arbitrary exponentiations of base a, having no knowledge of how to do so explicitly, but having knowledge of how to exponentiate base b, along with knowledge of the constant log_b(a).
There are two main change of base formulae for tetration:
\(
\begin{eqnarray}
{}^x a & = & {}^{\text{slog}_b({}^x a)} b \\
\\[5pt]
\\
{\Large ^{\normalsize x} a} & = & {\Large \lim_{n \to \infty}\log_a^{\circ n}\left({}^{\left(\normalsize n+x+\mu_b(a)\right)} b\right)}
\end{eqnarray}
\)
In either case, if it makes you feel better, you can explicitly state that x is a real > -2, or x is an integer >= -1, or whatever. And I've already stated that a and b should be greater than eta, though as a tool for fractionally iterating logarithms, it has applications with bases between 1 and eta.
The first formula, again, is a trivial restatement of the definition of slog_b(x) as the inverse of b^^x. The second displays some fundamental truth about the relationship of tetration in various bases, which isn't at all obvious by looking solely at the first formula, and it also allows us solve tetration for base a when we have no knowledge of how to do so explicitly, so long as we know how to do so with base b, and we know the value of the constant mu_b(a).
For real numbers, my formula is valid, assuming you define tetration as iterated exponentials, with negative iterations equivalent to logarithms. I don't claim validity over complex numbers, mainly because the formula's very basis for convergence is the increasing modulus of the successive tetrations, and complex tetrations can cause the modulus to decrease, which invalidates the limit.
For real x, use whatever domain you use for tetration. If *you* define tetration as valid for x>-2, what did you expect to happen when you put in x <= -2? The problem isn't with my formula, it's with your indecision on what the valid domain for x is. Use whatever domain you consider valid.
Moving along:
There are two main base conversion formulae for exponentiation for base a, given an exact solution for base b:
\(
\begin{eqnarray}
a^x & = & b^{\log_b(a^x)} \\
\\[5pt]
\\
a^x & = & b^{\log_b(a)\times x} \\
\end{eqnarray}
\)
The first is a trivial restatement of the definition of log_b(z) as the inverse function of b^z. The second displays some "fundamental truth" about exponentiation that isn't obvious from looking solely at the first formula. It allows you to solve for arbitrary exponentiations of base a, having no knowledge of how to do so explicitly, but having knowledge of how to exponentiate base b, along with knowledge of the constant log_b(a).
There are two main change of base formulae for tetration:
\(
\begin{eqnarray}
{}^x a & = & {}^{\text{slog}_b({}^x a)} b \\
\\[5pt]
\\
{\Large ^{\normalsize x} a} & = & {\Large \lim_{n \to \infty}\log_a^{\circ n}\left({}^{\left(\normalsize n+x+\mu_b(a)\right)} b\right)}
\end{eqnarray}
\)
In either case, if it makes you feel better, you can explicitly state that x is a real > -2, or x is an integer >= -1, or whatever. And I've already stated that a and b should be greater than eta, though as a tool for fractionally iterating logarithms, it has applications with bases between 1 and eta.
The first formula, again, is a trivial restatement of the definition of slog_b(x) as the inverse of b^^x. The second displays some fundamental truth about the relationship of tetration in various bases, which isn't at all obvious by looking solely at the first formula, and it also allows us solve tetration for base a when we have no knowledge of how to do so explicitly, so long as we know how to do so with base b, and we know the value of the constant mu_b(a).
~ Jay Daniel Fox
