Logarithm Calculator

1. Log mode. Enter any positive number to find its logarithm. Results are shown simultaneously for base 10 (log), base e (ln), and base 2 (log₂), plus a custom base of your choice. The number must be positive; logarithms of zero and negative numbers are undefined in real numbers.
2. Antilog mode. Enter an exponent to find the original number. This is the inverse operation: if log₁₀(x) = 3, then antilog₁₀(3) = 10³ = 1000. Useful for reversing scientific notation calculations.
3. Custom base. Enter any positive number (except 1) as the base for a custom logarithm. The calculator uses the change-of-base formula: log_b(x) = ln(x) / ln(b).

Quick check: log₁₀(100) = 2 because 10² = 100. ln(1) = 0 because e⁰ = 1. log₂(8) = 3 because 2³ = 8.

Definition:         log_b(x) = y  means  b^y = x

Product rule:       log(a × b) = log(a) + log(b)
Quotient rule:      log(a / b) = log(a) - log(b)
Power rule:         log(aⁿ) = n × log(a)
Change of base:     log_b(x) = log(x) / log(b)

Special values:
log_b(1) = 0   (any base)
log_b(b) = 1   (any base)
log_b(0) = undefined
ln(e) = 1      (natural log)

Common logarithms and their uses: