Find the Power Tower Height
A tetration tower repeatedly raises its base to the power of itself. This calculator searches for an integer height h satisfying a ↑↑ h = target.
What Is Tetration Height?
Tetration is an operation that repeatedly applies exponentiation. The height tells you how many copies of the base appear in the power tower.
2 ↑↑ 2 = 22 = 4
2 ↑↑ 3 = 2(22) = 16
2 ↑↑ 4 = 2(2(22)) = 65,536
How Does the Height Calculator Work?
The calculator starts with a tower of height 1, whose value is simply the base. It then adds one level at a time, using the previous tower value as the new exponent.
- Enter a base greater than 1.
- Enter the target number you want the tower to reach.
- The calculator evaluates successive integer tower heights.
- Each calculated value is compared with your target.
- The calculator reports a matching height or the closest computed height.
Why Must the Target Be at Least the Base?
For a base greater than 1, the first tower value is the base itself. Subsequent finite tower values are larger. Therefore, a target smaller than the base cannot be reached by a positive integer tetration height under this definition.
Can Tetration Height Be Fractional?
This calculator searches for positive integer heights. Fractional tetration is a more advanced topic: extending tetration to non-integer heights requires choosing a mathematical interpolation or fractional-tetration function. Different definitions can have different properties, so a fractional height is not uniquely determined by ordinary integer tetration alone.
Why Are Very Large Towers Difficult to Calculate?
Power towers grow extremely quickly for many bases. A tower can exceed the range of ordinary computer floating-point numbers after only a few levels. This calculator detects numerical overflow and reports the closest result it was able to calculate.
For bases between 1 and e1/e, the infinite tower has a finite limit. As a result, some targets may be larger than any finite tower value and cannot be reached. The calculator also detects when floating-point precision prevents further numerical progress.
Related Math Calculators
Continue exploring tetration, hyperoperations, and large-number calculations with these related tools.