Logarithm Calculator (log, ln, custom base)
Compute logarithms in any base — common (log₁₀), natural (ln), binary (log₂), or custom.
What this calculates
A logarithm answers 'what power of base gives this number?'. log₁₀(1000) = 3 because 10³ = 1000. Logarithms turn multiplication into addition and exponents into multiplication, which is why they're everywhere in math, science, and engineering.
Formula & how it works
log_b(x) = ln(x) / ln(b). Special bases: log₁₀ (common), ln = log_e (natural, e ≈ 2.71828), log₂ (binary).
Worked example
log₁₀(1000) = 3. ln(e²) = 2. log₂(64) = 6.
Frequently asked questions
Why use ln vs log?
ln (natural log) is calculus-friendly because d/dx[ln x] = 1/x. Common log (base 10) is convenient for scientific notation and pH/decibel scales.
Log of zero or negative?
Undefined for real numbers. log(0) → −∞, log(negative) needs complex numbers.
Change of base?
log_b(x) = log_a(x) / log_a(b) for any positive base a. Use this to compute any base from log or ln on a basic calculator.
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