Logarithm Calculator
Calculate log base 10, natural log (ln), log base 2, or any custom base logarithm. Instant results for all common log functions.
Logarithm Result
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Natural Log (ln) —
Log Base 10 —
Log Base 2 —
Extended More scenarios, charts & detailed breakdown ▾
log₁₀
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ln (base e) —
log₂ —
log₃ —
log₅ —
Professional Full parameters & maximum detail ▾
Individual Logs
Log of Each Number —
Min Log Value —
Max Log Value —
Combined
Sum of Logs (= log of product) —
Antilog of Sum (= product of inputs) —
How to Use This Calculator
Enter a positive Number and select the Log Base (10, e, 2, or custom). If you choose Custom Base, enter the base value. The calculator shows the result for your chosen base plus log₁₀, ln, and log₂ all at once.
Formula
log_b(x) = ln(x) / ln(b) • log(1000) = 3 • ln(e²) = 2
Example
log₁₀(1000) = 3, ln(e) = 1, log₂(8) = 3, log₃(81) = 4
Frequently Asked Questions
- A logarithm answers the question: "What exponent do I raise the base to in order to get this number?" The notation log_b(x) = y means b^y = x. For example, log₁₀(1000) = 3 because 10³ = 1000. Logarithms are the inverse of exponentiation, just as division is the inverse of multiplication. They compress large ranges of values, which is why they are used in the Richter scale (earthquakes), decibels (sound), pH (acidity), and virtually all of information theory and signal processing.
- The natural logarithm uses base e ≈ 2.71828 (Euler's number). ln(x) = log_e(x). Because e arises naturally in calculus as the base of exponential growth, ln appears in derivatives, integrals, compound interest, population growth, radioactive decay, and statistical distributions. Key values: ln(1) = 0, ln(e) = 1, ln(e²) = 2. The natural log and the exponential function e^x are inverses: ln(e^x) = x and e^(ln x) = x.
- Log base 2 (log₂) answers: "How many times must I double to reach x?" For example, log₂(8) = 3 because 2³ = 8, and log₂(1024) = 10 because 2¹⁰ = 1024. Log base 2 is fundamental in computer science because computers store data in binary. It tells you the number of bits needed to represent x distinct values. Information theory uses log₂ to measure entropy in bits. Also: log₂(x) = ln(x) / ln(2) ≈ ln(x) / 0.6931.
- The change of base formula converts a logarithm in any base to one using natural log or log base 10: log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b). This is essential because most calculators only have ln and log₁₀ buttons. For example, log₃(81) = ln(81) / ln(3) = 4.394 / 1.099 = 4 (since 3⁴ = 81). Using this formula you can compute log₅(200), log₇(343), or any custom base logarithm.
- The number x must be strictly positive (x > 0) — the logarithm of zero or any negative number is undefined in real arithmetic. For the change of base or custom base calculation, the base b must also be positive and not equal to 1 (log base 1 is undefined because 1^y = 1 for all y, never reaching any other value). If you enter a negative or zero number, the calculator will return an error. Logarithms of numbers between 0 and 1 are negative: log₁₀(0.01) = −2.