Study repeated logs for algorithmic growth behavior. See iterations, export results, and compare stopping rules. Built for quick checks, examples, graphs, and structured explanations.
| n | Base | Threshold | Iterated log | Notes |
|---|---|---|---|---|
| 16 | 2 | 1 | 3 | 16 → 4 → 2 → 1 |
| 65536 | 2 | 1 | 4 | Large binary value with a very small log* result |
| 1000000 | 10 | 1 | 2 | Useful for powers of ten |
| 1000000000000 | 2.718281828 | 1 | 4 | Natural logarithm version for continuous models |
The calculator applies a logarithm repeatedly until the running value becomes less than or equal to the chosen threshold.
L₀(n) = n Lₖ(n) = log_b(Lₖ₋₁(n)) log*_b(n; t) = smallest k such that Lₖ(n) ≤ t
Here, n is the starting value, b is the base, and t is the stopping threshold. The default interpretation usually uses t = 1.
Iterated logarithms grow slower than ordinary logarithms. That makes them useful when you want to describe procedures that collapse large numbers very quickly after repeated shrinking steps.
In theoretical machine learning, algorithm design, and data structure analysis, log* terms appear when costs rise so slowly that even huge inputs still produce tiny values. This makes the measure helpful for reasoning about scalable procedures and asymptotic behavior.
This calculator shows that behavior directly. You can inspect each reduction step, compare different bases, adjust the stopping threshold, and export the sequence for notes, reports, or classroom discussion.
An iterated logarithm counts how many times you apply a logarithm before the value becomes small enough, usually less than or equal to 1.
The base changes each reduction step. A larger base usually shrinks values faster, so the final log* count can become smaller.
Many definitions of log* stop once the running value reaches 1 or less. It gives a standard benchmark that is easy to compare across examples.
Yes. If your starting value is already at or below the threshold, the calculator returns 0 iterations because no logarithmic reduction is needed.
No. Iterated log repeatedly applies logarithms. Exponent rules describe how powers combine, which is a different operation and a different growth pattern.
It appears mainly in theory, complexity analysis, and supporting algorithm discussions. It helps describe processes whose cost grows so slowly that the result stays tiny for enormous inputs.
Repeated logarithms crush large values extremely fast. Even very large inputs often need only a few iterations before crossing the stopping threshold.
Yes. Set the base to 2.718281828 to model repeated natural logarithms for continuous or analytic work.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.