Super Logarithm Calculator

Calculate the inverse of tetration using the super logarithm, commonly written as slog. Enter a base and target value to find the corresponding tower height.

Calculate Super Logarithm

Base (a)

Use a real base greater than 1.

Target Value (x)

The target must be at least 1.

Tolerance

Controls how close a tower value must be to count as a match.

Maximum Tower Height

Limits the number of tower levels evaluated.

What Is a Super Logarithm?

A super logarithm is an inverse operation of tetration. Just as an ordinary logarithm reverses exponentiation, a super logarithm reverses repeated exponentiation.

If a number is created by repeatedly raising a base to the previous result, the super logarithm describes the number of tower levels needed to obtain that value.

T₀ = 1
T₁ = a
T₂ = a^a
T₃ = a^{a^a}

slog_a(T_n) = n

Here, the height-zero convention is T₀ = 1. This makes the first positive integer height equal to the base itself.

Super Logarithm vs. Ordinary Logarithm

An ordinary logarithm answers the question: "What exponent produces this number?" A super logarithm asks a higher-level question: "How many repeated exponentiation levels produce this number?"

Ordinary logarithm:
log_a(x) = y means a^y = x

Super logarithm:
slog_a(x) = h means T_h = x

The super logarithm is therefore useful when studying tetration, power towers, hyperoperations, and extremely fast-growing functions.

How This Calculator Works

It starts with the height-zero tower value, T₀ = 1.
It repeatedly calculates the next tower value using the selected base.
It compares each tower value with the target.
If a tower value matches within tolerance, it reports that integer height.
If the target lies between two tower values, it estimates a fractional height through logarithmic interpolation.

Calculations use PHP floating-point numbers. Extremely large towers may exceed the available numerical range. Fractional-height results should be treated as interpolation estimates rather than exact values from a particular analytic fractional-tetration construction.

Frequently Asked Questions

1. What does slog mean?

Slog is an abbreviation for super logarithm. It represents an inverse operation of tetration and describes the height of a power tower for a given base and target.

2. Is the super logarithm the inverse of tetration?

Yes. For integer heights under the convention used here, the super logarithm reverses tetration. Fractional heights require an extension of tetration and are not uniquely determined by integer-height values alone.

3. Can a super logarithm have a fractional result?

Fractional super logarithms can be studied using continuous extensions of tetration. This calculator provides a numerical interpolation estimate between consecutive integer tower heights, not a full analytic fractional tetration solution.

4. Why must the target be at least 1?

This calculator uses positive real power towers with T₀ = 1 and a base greater than 1. Its supported tower values therefore start at 1 and increase.

5. What is the relationship between tetration and super logarithms?

Tetration repeatedly applies exponentiation. The super logarithm reverses that process by determining the tower height associated with a value.