The logarithm log_b(x) is the exponent to which you must raise b to get x. log₂(8) = 3 because 2³ = 8.

What is the difference between log and ln?

log usually means log₁₀ (common logarithm). ln means the natural logarithm with base e ≈ 2.71828.

Why is log undefined for negative numbers?

No real power of a positive base produces a negative result. Logarithms of negative numbers are complex and outside this tool's scope.

Enter a positive number to compute log₁₀, ln, and log₂ simultaneously — or switch to custom base mode to calculate log_b(x) for any valid base b.

Logarithm definitions and change of base

log₁₀(x) = log(x)/log(10) · ln(x) = loge(x) · log_b(x) = ln(x)/ln(b)

Logarithm is defined only for x > 0 and base b > 0, b ≠ 1.

x = 100: log₁₀(100) = 2, ln(100) ≈ 4.605, log₂(100) ≈ 6.644. Custom base 5: log₅(100) = ln(100)/ln(5) ≈ 2.861.

log₁₀(1000) = 3 because 10³ = 1000 — the common log counts the number of digits minus 1.
ln(e) = 1 where e ≈ 2.71828 (Euler's number). The natural log is the inverse of exponential growth.
log₂ is fundamental in information theory — log₂(1024) = 10 bits.
Change of base: log_b(x) = ln(x) / ln(b) — any logarithm can be computed from the natural log.

Used in pH (chemistry), Richter magnitude (earthquakes), and decibel (sound). log₁₀(x) answers '10 to what power equals x?'

Used in calculus, probability, and finance (continuous compounding). It's the inverse of eˣ.

Fundamental in computer science: the number of bits needed to represent n values is ⌈log₂(n)⌉.

What is a logarithm?: The logarithm log_b(x) is the exponent to which you must raise b to get x. log₂(8) = 3 because 2³ = 8.
What is the difference between log and ln?: log usually means log₁₀ (common logarithm). ln means the natural logarithm with base e ≈ 2.71828.
Why is log undefined for negative numbers?: No real power of a positive base produces a negative result. Logarithms of negative numbers are complex and outside this tool's scope.

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