Calculate Inverse Tetration
The super-logarithm is the inverse operation of tetration. Given a base a and a target x, this calculator investigates the height h satisfying a ↑↑ h = x.
What Is Inverse Tetration?
Tetration is repeated exponentiation. Inverse tetration asks how many levels of exponentiation are needed to produce a particular value.
sloga(x) = h
The function slog is called the super-logarithm. It is analogous to the ordinary logarithm, which reverses exponentiation. The super-logarithm reverses tetration.
Example: Base 2
2 ↑↑ 2 = 4
2 ↑↑ 3 = 16
2 ↑↑ 4 = 65,536
Therefore:
Super-Logarithm and Ordinary Logarithm
The ordinary logarithm reverses exponentiation:
The super-logarithm reverses tetration:
Integer and Fractional Heights
Integer tetration heights have a direct definition as finite power towers. Fractional heights require an extension of tetration between integer levels.
There is no single universally mandatory fractional-tetration convention. This calculator therefore reports exact integer matches separately and uses linear interpolation only as an explicitly labelled estimate for targets between adjacent integer tower values.
Numerical Limitations
Power towers grow very quickly for many bases. Ordinary floating-point arithmetic has a finite numerical range and precision. Very tall towers may overflow, and values that are extremely close together may become indistinguishable.
The calculator is intended for supported positive bases greater than 1 and finite positive target values. Its numerical interpolation should not be confused with a rigorously defined analytic super-logarithm.
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