User:Lemio/sandbox

Over-Exponential functions
Over-exponentiation is a mathematical operation, written as $$b^{n\uparrow m}$$, involving three numbers, the base b, exponent (or power) n and the over-exponent (or over-power) m. When m is a positive integer, over-exponentiation corresponds to repeated exponentiation.
 * $$b^{n\uparrow 0} = b$$
 * $$b^{n\uparrow 1} = b^{n}$$
 * $$b^{n\uparrow 2} = (b^{n})^n$$
 * $$b^{n\uparrow 3} = ((b^{n})^n)^n$$
 * $$\cdots$$

just as exponentiation by a positive integer corresponds to repeated multiplication:
 * $$b^n = \underbrace{b \times \cdots \times b}_n,$$

A formula to calculate $$b^{n\uparrow m}$$ when m is not a positive integer is:
 * $$b^{n\uparrow m} = e^{n^m*ln(b)}$$

Prove

 * $$b^{n\uparrow m} = b^{n\uparrow m}$$
 * $$ln(b^{n\uparrow m}) = ln(b^{n\uparrow m})$$
 * $$ln(b^{n\uparrow m}) = n^m*ln(b)$$
 * $$b^{n\uparrow m} = e^{n^m*ln(b)}$$
 * $$b^{n\uparrow m} = e^{n^m*ln(b)}$$

The inverse function

 * $$v=b^{n\uparrow m}$$
 * If you want to know v:
 * $$v=e^{n^m*ln(b)}$$
 * If you want to know m
 * $$m=log_n(log_b(m))$$
 * If you want to know n
 * $$n=(log_b(v))^{\frac{1}{m}}$$
 * If you want to know b
 * $$b=e{\frac{ln(v)}{n^m}}$$
 * $$x=e^{n^y*ln(b)}$$
 * $$ln(x)=n^y*ln(b)$$
 * $$\frac{ln(x)}{ln(b)}=n^y$$
 * $$y=log_n(log_b(x))$$

Tests
$$({10^{10^{Q-3}}})^{100^x}$$