Inverse Logarithm Calculators

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The inverse logarithm — also called the antilogarithm (antilog) — reverses the logarithm operation. If log₁₀(x) = y, then the antilog base 10 of y is 10^y = x. Similarly, if ln(x) = y, then the natural antilog is e^y = x. Antilogs are essential in science for converting log-scale values back to linear values — pH to hydrogen ion concentration, Ct values to qPCR copy numbers, optical density to cell density, and log-transformed data back to original units.

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Antilog Base 10

If log₁₀(x) = y, then x = 10^y.

  • log₁₀(1000) = 3 → antilog(3) = 10³ = 1000
  • log₁₀(0.001) = −3 → antilog(−3) = 10⁻³ = 0.001
  • log₁₀(500) ≈ 2.699 → antilog(2.699) = 10^2.699 ≈ 500

On a calculator: enter the value, press 10^x (or INV → LOG).

Natural Antilog (e^x)

If ln(x) = y, then x = e^y, where e ≈ 2.71828. On a calculator: enter the value, press e^x (or INV → LN). In spreadsheets: =EXP(value) for natural antilog; =10^(value) for base-10 antilog.

pH to [H⁺]: Classic Antilog Application

[H⁺] = 10^(−pH). For pH 4.5: [H⁺] = 10^(−4.5) = 3.16 × 10⁻⁵ M. Each full pH unit change represents a 10-fold change in [H⁺] — a log-to-linear conversion via antilog.

qPCR Antilog Application

In qPCR standard curves, log(copy number) is plotted vs. Ct. To convert Ct to copy number: derive log(copy number) from the regression equation, then: copy number = 10^(log copy number). This is done routinely when quantifying gene expression or viral load from Ct values.

Glossary

Antilogarithm (Antilog)
The inverse operation of a logarithm; antilog base 10 of y = 10^y; antilog base e of y = e^y; converts log-scale values back to linear values.
Base-10 Logarithm (log₁₀)
The power to which 10 must be raised to produce a given number; its inverse (antilog) is 10^x; used in pH, spectrophotometry, and qPCR calculations.
Natural Logarithm (ln)
The logarithm to base e (≈ 2.71828); its inverse is e^x; used in growth rate equations, thermodynamics, and statistical transformations.

Frequently Asked Questions

An antilog (inverse logarithm) reverses a logarithm. For base 10: antilog₁₀(y) = 10^y. For natural log: antiln(y) = e^y. Example: if log₁₀(x) = 3.7, then x = 10^3.7 ≈ 5012. On a scientific calculator, use the 10^x key for base-10, or e^x for natural antilog. In Excel or Google Sheets: =10^(value) or =EXP(value) respectively.

Since pH = −log₁₀[H⁺], taking the antilog gives: [H⁺] = 10^(−pH). For pH 3: [H⁺] = 10^(−3) = 0.001 M. For blood pH 7.4: [H⁺] = 10^(−7.4) ≈ 3.98 × 10⁻⁸ M. A drop from pH 7.4 to pH 7.0 increases [H⁺] by 2.51-fold — calculated as antilog of the pH difference: 10^0.4 ≈ 2.51.

Log asks: what power must the base be raised to in order to produce this number? log₁₀(1000) = 3 because 10³ = 1000. Antilog asks: if the log is this value, what is the original number? antilog₁₀(3) = 10³ = 1000. They are inverse operations — applying log then antilog returns the original number, just like squaring and taking a square root.

Common antilog uses in biology: converting pH to [H⁺] (10^−pH); converting qPCR standard curve output (log copy number) to actual copy numbers (10^log_copies); back-transforming log-normalized gene expression data to fold-change; converting optical density to cell density via a calibration curve; and recovering geometric means from log-transformed population data.