Calculate Repeated Logarithms
Enter a positive starting number, choose a logarithm base, and specify how many times the logarithm should be applied.
What Is an Iterated Logarithm?
An iterated logarithm is obtained by applying the logarithm function repeatedly to a starting number. Each new iteration uses the previous result as its input.
x₁ = logb(x₀)
x₂ = logb(x₁)
x₃ = logb(x₂)
xₙ₊₁ = logb(xₙ)
Here, N is the starting number and b is the logarithm base. The calculator repeats this operation until it reaches a stopping condition or the maximum number of iterations.
Example: Repeated Base-10 Logarithms
Starting with 1,000,000 and applying base-10 logarithms:
x₁ = log₁₀(1,000,000) = 6
x₂ = log₁₀(6) ≈ 0.778151
x₃ = log₁₀(0.778151) ≈ -0.1089
The next real logarithm cannot be calculated because the input is negative. The real-valued iteration therefore stops at this point.
Important Mathematical Properties
Domain of the logarithm
A real logarithm is defined only for a positive input. If an iteration produces zero or a negative number, another real logarithm cannot be applied.
The fixed point at 1
For every valid logarithm base, logb(1) = 0, while logb(b) = 1. A value of 1 is not a fixed point of the logarithm function: applying another logarithm gives 0.
Base selection
The calculator accepts bases greater than zero except 1. Bases greater than 1 and bases between 0 and 1 are both mathematically valid, although their iteration behavior can differ substantially.
Stopping tolerance
The tolerance is used to detect when successive values are nearly equal. It does not change the mathematical definition of the logarithm or guarantee convergence.
Frequently Asked Questions
1. What does an iterated logarithm calculate?
It repeatedly applies a logarithm to the previous result. For example, the second iteration is the logarithm of the first logarithm, using the same base.
2. Can I use a base other than 10?
Yes. You can use any positive base except 1, including 2, 10, and values between 0 and 1.
3. Why does the calculation sometimes stop early?
A real logarithm requires a positive input. If an iteration produces zero or a negative number, the next real logarithm is undefined, so the calculator stops.
4. Does an iterated logarithm always converge?
No. Depending on the starting number and base, the values can become non-positive, stabilize, or continue changing. The calculator reports the actual stopping condition.
5. What is the difference between a logarithm and an iterated logarithm?
A standard logarithm is applied once. An iterated logarithm applies the same logarithm repeatedly, using each result as the input for the next calculation.
Related Math Calculators
Explore other calculators for logarithms, exponential functions, tetration, and higher hyperoperations.