Compute a nested logarithm of the form logb₂(logb₁(x)). Enter your value, choose bases, and click calculate.
What is a log log calculation?
A “log log” expression means you apply a logarithm twice. First, you compute an inner logarithm, and then you take the logarithm of that result. In symbols:
This calculator gives you a quick and accurate way to evaluate nested logarithms with any valid bases.
Formula used by this calculator
The calculator uses the change-of-base identity:
So the full computation is:
result = ln(inner) / ln(b₂)
Domain rules (very important)
- x > 0 (you cannot take a log of zero or a negative number)
- b₁ > 0 and b₁ ≠ 1 (valid logarithm base rule)
- inner = logb₁(x) > 0 (outer log input must be positive)
- b₂ > 0 and b₂ ≠ 1
How to use the log log calculator
- Enter your value for x.
- Choose the inner base (b₁).
- Choose the outer base (b₂).
- Set decimal precision and click Calculate.
The result panel also shows step-by-step intermediate values so you can verify your work.
Worked examples
Example 1: log10(log10(100))
First compute inner log: log10(100) = 2. Then outer log: log10(2) ≈ 0.30103.
Example 2: log2(log2(256))
Inner: log2(256) = 8. Outer: log2(8) = 3. Final answer: 3.
Example 3: ln(ln(e4))
If both bases are e, then this is a natural log of a natural log. Inner ln(e4) = 4, outer ln(4) ≈ 1.38629.
Where log log expressions appear
- Computer science: complexity analysis for very slowly growing functions.
- Mathematical modeling: scale compression of already logarithmic quantities.
- Theory and proofs: asymptotics and bounds in number theory and algorithms.
Common mistakes to avoid
- Using a base of 1 (invalid for logarithms).
- Assuming any positive x works; the outer log still needs a positive inner result.
- Confusing log(log(x)) with (log(x))2; these are completely different.
- Forgetting whether your class or software expects base 10 or base e by default.
Quick takeaway
A log log calculator helps you evaluate nested logarithms accurately and quickly, especially when bases differ. Use the tool above, check domain constraints, and review the step-by-step output to build confidence in your result.