What Is a Logarithm Calculator?
A logarithm calculator finds the exponent needed to raise a base to produce a given number. This tool calculates a custom-base logarithm, plus the three most common logarithms: base 10, natural (base e), and base 2.
How to Use the Log Calculator
- Enter the value (x) you want the logarithm of.
- Enter the base.
- View the result, plus common log, natural log, and binary log for reference.
Logarithm Formula
Worked Example
log₁₀(100) = 2, because 10² = 100
log₂(8) = 3, because 2³ = 8
Understanding Your Results
A logarithm answers "what power do I need to raise the base to, to get this number?" Common log (base 10) is often used in science and engineering. Natural log (base e ≈ 2.71828) appears throughout calculus and growth/decay models. Binary log (base 2) is common in computer science.
Logarithm Rules Reference
| Rule | Formula |
|---|---|
| Product Rule | log(xy) = log(x) + log(y) |
| Quotient Rule | log(x/y) = log(x) − log(y) |
| Power Rule | log(xⁿ) = n × log(x) |
Common Mistakes to Avoid
- Attempting to take the logarithm of zero or a negative number, which is undefined for real numbers.
- Confusing natural log (ln) with common log (log base 10) — they're different bases with different results.
- Forgetting the base when writing or interpreting a logarithm — log(x) alone is ambiguous without context.
Frequently Asked Questions
A logarithm answers the question: what exponent do I need to raise a given base to, in order to get a specific number? For example, log₁₀(100) = 2 because 10² = 100.
"log" typically refers to base-10 logarithm (common log), while "ln" refers to natural logarithm (base e, approximately 2.71828).
Not for real numbers — logarithms of zero or negative numbers are undefined in the real number system, though they exist in complex number systems.
Use the change of base formula: log_b(x) = ln(x) / ln(b), which lets you compute any base's logarithm using natural log.
Measuring earthquake magnitude (Richter scale), sound intensity (decibels), pH levels in chemistry, and computer science algorithm complexity all use logarithmic scales.