Logarithm calculator

ln is the natural logarithm, with base e ≈ 2.718282, while log on most calculators means base 10. Some textbooks and programming languages use log for the natural logarithm instead; Python's math.log(10) returns 2.302585. The two differ by a constant factor, ln x = ln 10 × log₁₀ x ≈ 2.302585 × log₁₀ x, so ln 10 ≈ 2.302585 while log₁₀ 10 = 1.

Use the change-of-base rule: log_b x = ln x ÷ ln b, or equally log₁₀ x ÷ log₁₀ b. For log₂ 1024 that is 6.931472 ÷ 0.693147 = 10. For log₄ 8 it gives 1.5, which is exact because 4^(3/2) = (√4)³ = 8. Any base works except 1, and the base must be positive.

Take logarithms of both sides: x × ln 3 = ln 20, so x = ln 20 ÷ ln 3 ≈ 2.995732 ÷ 1.098612 ≈ 2.726833. The same method finds doubling times: money growing 7% a year doubles when 1.07ˣ = 2, at x = ln 2 ÷ ln 1.07 ≈ 10.24 years.

No real power of a positive base gives 0 or a negative number: 2ˣ is positive for every real x and only approaches 0 as x heads toward −∞. So log₂ 0 has no value and ln(−1) has no real value; in complex numbers ln(−1) = iπ. A base of 1 is excluded too, since 1ˣ = 1 for every x.

log₂ x counts how many times 1 must be doubled to reach x. log₂ 1024 = 10, so 1,024 = 2¹⁰ and whole numbers from 0 to 1,023 fit in 10 bits. It also bounds halving processes: a binary search of 1,024 sorted items needs at most ⌊log₂ 1024⌋ + 1 = 11 comparisons.

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 6 worked examples whose answers come from independent sources; for example, “log₂ 1024” is checked against 2¹⁰ = 1024.

NIST Digital Library of Mathematical Functions §4.2 — logarithms, change of base (4.2.E17); Khan Academy — Change of base formula for logarithms.