📊 Logarithm Calculator

log₁₀(1000) = 3 because 10³ = 1000
ln(e²) = 2 because e² = e²
log₂(8) = 3 because 2³ = 8

log(a × b) = log(a) + log(b)

Example: log(6) = log(2×3) = log(2) + log(3)
= 0.3010 + 0.4771 = 0.7781

log(a / b) = log(a) − log(b)

Example: log(5) = log(10/2) = log(10) − log(2)
= 1 − 0.3010 = 0.6990

log(aⁿ) = n × log(a)

Example: log(1000) = log(10³) = 3 × log(10) = 3

logₙ(x) = log(x) / log(n) = ln(x) / ln(n)

Example: log₅(25) = log(25)/log(5) = 1.3979/0.6990 = 2
Verify: 5² = 25 ✓

log(1) = 0 (any base, because b⁰ = 1)
log(b) = 1 (log of the base itself = 1)
log(0) = undefined (−∞ approaching)
log(−x) = undefined (in real numbers)

ln(e) = 1 (e ≈ 2.71828...)
ln(1) = 0
log₂(2) = 1

Antilog₁₀(x) = 10ˣ
Antilog(ln) = eˣ
Antilog₂(x) = 2ˣ

Example: antilog₁₀(3) = 10³ = 1000
If log(x) = 2.5, then x = 10^2.5 = 316.23

Richter Scale: M = log₁₀(A/A₀) (earthquake)
Decibels: dB = 10×log₁₀(P₁/P₀) (sound)
pH scale: pH = −log₁₀([H⁺]) (acidity)
Compound int: t = ln(A/P) / (r) (finance)
Binary search: steps = log₂(n) (computing)
Info theory: bits = log₂(1/p) (entropy)