Calculation details
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What does this calculator estimate?
A logarithm calculator finds the exponent needed to reach a number for a given base — natural (ln), base 10, or base 2. Enter a positive number and choose the base to see its logarithm.
- logₐ(x) answers: what power of a gives x?
- ln = base e, log₁₀ = base 10, log₂ = base 2
- Only defined for positive numbers
What a logarithm means
A logarithm is the inverse of exponentiation: logₐ(x) asks 'to what power must I raise a to get x?' Natural logs (base e ≈ 2.718) arise in growth and decay; base 10 in orders of magnitude; base 2 in computing and information theory.
Limitations to watch for
Logarithms are only defined for positive inputs — log of zero or a negative number is undefined. The output can be negative (for x between 0 and 1). Rounding at six decimals is applied. Changing the base changes the value: log₂(8) = 3 but log₁₀(8) ≈ 0.903.
How to use it in practice
Use logs to linearize exponential data, measure growth rates, decode decibels and pH (both logarithmic scales), and analyze algorithm complexity. Remember the change-of-base rule to convert between bases.
['Enter a positive number.', 'Choose the base: natural (ln), 10, or 2.', 'The tool returns the exponent for that base.']
What logarithms are
A logarithm answers 'what exponent gives this number': log₁₀(100) = 2 because 10² = 100. The calculator evaluates logs in common (base 10), natural (base e), and custom bases.
The base matters
log₁₀ is the common log (pH, decibels, Richter); ln is the natural log (growth, compound interest); log₂ is the binary log (information, algorithms). The same number gives different results across bases — the calculator handles all three.
Log properties
log(ab) = log(a) + log(b), log(a/b) = log(a) − log(b), log(a^n) = n·log(a). These properties turn multiplication into addition — the reason logs power slide rules and remain useful in data work.
A worked example
log₁₀(1,000) = 3; ln(e³) = 3; log₂(32) = 5. The calculator returns the value for any base from the inputs.
Real-world logs
pH is −log₁₀[H+]; decibels are 10·log₁₀(power ratio); doubling times use ln. The calculator is the quick evaluator behind all of them.
How this calculator works
Formula
logₐ(x) = y means a^y = x. The tool computes ln(x) (natural log, base e), log₁₀(x) (common log), and log₂(x) (binary log). Input must be positive.
Worked example
log₁₀(1000) = 3 because 10³ = 1000. log₂(8) = 3 because 2³ = 8. ln(e) = 1.
Assumptions to verify
- The input is a positive real number.
- Standard definitions of ln, log₁₀, and log₂ are used.
- Results are rounded to six decimal places.
Frequently asked questions
What is a logarithm?
The inverse of exponentiation. logₐ(x) asks: to what power must I raise the base a to get x? For example, log₁₀(1000) = 3.
What is the natural logarithm (ln)?
A logarithm with base e (≈ 2.718). It's the default in calculus, growth models, and finance: ln(e) = 1.
What is log₂ used for?
Binary logarithms are everywhere in computing — information bits, binary search, and algorithm complexity (log₂ of 8 is 3, meaning 2³ = 8).
Why must the input be positive?
Because a power of a positive base is always positive — you can never reach zero or a negative number by raising a positive base to a real exponent.
What does a negative logarithm mean?
The number is between 0 and 1. For example, log₁₀(0.01) = −2 because 10⁻² = 0.01.
How do I convert between bases?
Use the change-of-base rule: logₐ(x) = log_b(x) ÷ log_b(a). For example, log₂(8) = log₁₀(8) ÷ log₁₀(2) ≈ 3.
Where do logarithms appear in real life?
pH, decibels, earthquake magnitude, sound levels, compound growth, doubling times, and computer science complexity.
What is the difference between log and ln?
Log is base 10; ln is base e (natural).
What is the log of a negative number?
Undefined in real numbers — logs require positive arguments.
Cite this tool
BoringToolsKit. “Logarithm Calculator.” boringtoolskit.com/log-calculator/ (reviewed 2026-08-25). Free to reference in articles, syllabi, and answer posts with a link.
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