Inverse Logarithm Calculator

The inverse logarithm of y in base b is the number x = b^y. It answers the question: "which number has a logarithm of y?"

Logarithms and exponentials undo each other. Since log₁₀ 1000 = 3, the antilog of 3 is 10³ = 1000. Since ln e² = 2, the natural antilog of 2 is e² ≈ 7.389.

1. Common antilog: base 10, written 10^y or antilog(y).
2. Natural antilog: base e ≈ 2.71828, written e^y or exp(y).
3. Binary antilog: base 2, written 2^y, used in computing and information theory.

Every whole step in y multiplies the answer by the base, and the fractional part supplies the leading digits.

Split y into a whole-number characteristic and a decimal mantissa between 0 and 1. Look up the antilog of the mantissa, then move the decimal point by the characteristic.

1. Characteristic: take the floor of y. For y = 2.4771 it is 2. For y = −1.3 it is −2, not −1.
2. Mantissa: subtract the characteristic. For 2.4771 it is 0.4771; for −1.3 it is 0.7.
3. Look up the digits: 10^0.4771 ≈ 3.000, found in an antilog table at row 0.47, column 7.
4. Place the point: multiply by 10^{characteristic}: 3.000 × 10² = 300.0.

Negative logs are often written in bar notation, such as 2.7 for −2 + 0.7 = −1.3. Its antilog is 5.012 × 10⁻² = 0.05012.

The natural antilog e^x turns a natural log back into a quantity, and it is the formula for continuous growth and decay: A = P·e^rt.

Taking ln of both sides gives ln(A/P) = rt, which is how you solve for a rate or a time. The inverse log then converts your answer back: if ln(A/P) = 0.6931, the growth factor is e^0.6931 = 2, a doubling.

• Finance: $1,000 at 5% compounded continuously for 10 years grows to 1000·e^0.5 ≈ $1,648.72.
• Radioactive decay: N = N₀·e^−λt, with half-life ln 2 / λ.
• Population: a 2% continuous growth rate doubles a population in ln 2 / 0.02 ≈ 34.7 years.

Many scientific scales are logarithms, so converting a reading back into a physical quantity is an inverse log.

• pH: pH = −log[H⁺], so [H⁺] = 10^−pH. Black coffee at pH 5 has 100 times more hydrogen ions than pure water at pH 7.
• Decibels: L = 10·log(I/I₀), so I/I₀ = 10^L/10. A 30 dB increase means 1,000 times the sound intensity.
• Earthquakes: energy grows by 10^1.5 ≈ 31.6 per magnitude step, so a magnitude 7 releases about 1,000 times the energy of a magnitude 5.
• Star brightness: a difference of m magnitudes is a brightness ratio of 10^0.4m.

Because b^{p + q} = b^p·b^q, adding two logs and taking the antilog multiplies the original numbers. This is the idea behind log tables and the slide rule, which engineers used for multiplication for more than 300 years.

• Product: antilog(p + q) = antilog(p) × antilog(q).
• Quotient: antilog(p − q) = antilog(p) ÷ antilog(q).
• Power: antilog(k·p) = antilog(p)^k.
• Root: antilog(p/n) = the n-th root of antilog(p).

To multiply 2.5 × 3.2 with logs: log 2.5 + log 3.2 = 0.39794 + 0.50515 = 0.90309, and antilog(0.90309) = 8.