What is the difference between antilog base 10 and natural antilog?

Antilog base 10 (common antilog) raises 10 to the given power: antilog_10(y) = 10^y. It is used in science, engineering, pH calculations, and the Richter scale. Natural antilog uses Euler's number e (≈ 2.71828): antilog_e(y) = e^y, also written as exp(y). It appears in calculus, continuous growth/decay models, and finance.

What is the antilog table and how to use it?

An antilog table gives pre-computed values of 10^y for values of y between 0 and 1 (the mantissa). To use it: (1) Write your log value as y = characteristic + mantissa (e.g., 2.5463 = 2 + 0.5463). (2) Look up the mantissa row (0.54) and column (6) in the antilog table to get a 4-digit number (e.g., 3519). (3) Add the mean difference for the last digit (3). (4) Result = (table value) × 10^(characteristic + 1) = 3519 × 10^(2+1-1) = 351.9... ≈ 352.

Antilog Calculator

Inverse logarithm — Base 10, Base e (natural), Base 2 & Custom — with step-by-step solution

Select Base

Log Value / Exponent (y)

antilog₁₀(y) = 10^y

Quick Examples

What is an Antilogarithm (Antilog)?

An antilogarithm (or antilog) is the inverse operation of a logarithm. While a logarithm answers "to what power must the base be raised to get this number?", an antilogarithm answers the reverse: "what number do I get when I raise the base to this power?"

Formally, if log_b(x) = y, then the antilog is x = b^y. This relationship is the foundation of every antilog calculation.

antilog_b(y) = b^y
where b > 0, b ≠ 1, and y is any real number

Types of Antilogarithms

Type	Base	Formula	Common Use
Common Antilog	10	antilog(y) = 10^y	Science, engineering, pH, Richter scale
Natural Antilog	e ≈ 2.71828	exp(y) = e^y	Calculus, finance, biology, physics
Binary Antilog	2	antilog₂(y) = 2^y	Computer science, information theory
Custom Base	Any b > 0, b ≠ 1	antilog_b(y) = b^y	Specialized scientific computation

How to Use This Antilog Calculator

Select your base: Base 10 (common), Base e (natural), Base 2, or enter a custom base.
Enter the log value (exponent y) — it can be positive, negative, or a decimal.
Click Calculate Antilog to see the result instantly.
Use quick-example buttons to explore common values in one click.
Expand Step-by-Step Solution to see the full working.

Worked Examples

Example 1 — Common Antilog (Base 10)

Find antilog₁₀(3).

antilog₁₀(3) = 10³ = 1,000

Verification: log₁₀(1000) = 3 ✓

Example 2 — Natural Antilog (Base e)

Find antilog_e(2) = e².

e² = 2.71828² ≈ 7.3891

Verification: ln(7.3891) ≈ 2 ✓

Example 3 — Negative Exponent

Find antilog₁₀(-2).

10⁻² = 1/100 = 0.01

Used in pH: if pH = 2, [H⁺] = 10⁻² = 0.01 mol/L ✓

Example 4 — Decimal Exponent

Find antilog₁₀(0.5).

10^0.5 = √10 ≈ 3.1623

Verification: log₁₀(3.1623) ≈ 0.5 ✓

Antilog with Characteristic & Mantissa (Log Table Method)

In traditional mathematics (and many school exams), antilog is computed using a printed antilog table. A log value like 2.5463 is split into two parts:

Characteristic = 2 (the integer part). Determines the power of 10.
Mantissa = 0.5463 (the positive decimal part). Looked up in the antilog table.

Using an antilog table for mantissa 0.5463: look up row 0.54, column 6, mean difference 3 → gives 3519. The antilog = 3519 × 10^(2+1−4) = 351.9. Use our Advanced tab above to replicate this instantly.

Real-World Applications of Antilog

Chemistry (pH): The concentration of hydrogen ions is [H⁺] = 10^(−pH). At pH 7, [H⁺] = 10⁻⁷ = 0.0000001 mol/L.
Seismology (Richter Scale): Earthquake energy increases as 10^(1.5 × magnitude). A magnitude-8 quake releases antilog₁₀(12) = 10¹² ergs relative units.
Finance (Compound Growth): Continuous compounding formula A = P × e^(rt) uses the natural antilog.
Sound (Decibels): Converting dB back to amplitude ratio: ratio = 10^(dB/20).
Computer Science: Binary antilog 2^n gives the number of unique values in n bits (e.g., 2^8 = 256 colours in 8-bit colour).
Biology (Population Growth): Population after time t: N = N₀ × e^(kt), where e^(kt) is the natural antilog.

Key Properties of Antilog

Property	Formula	Example
Inverse of log	antilog_b(log_b(x)) = x	antilog₁₀(log₁₀(100)) = 100
Product rule	b^(a+c) = b^a × b^c	10^(1+2) = 10 × 100 = 1000
Zero exponent	b^0 = 1	antilog₁₀(0) = 1
Negative exponent	b^(-y) = 1/b^y	antilog₁₀(-3) = 0.001
Fractional exponent	b^(1/n) = n-th root of b	antilog₁₀(0.5) = √10 ≈ 3.162

Frequently Asked Questions

What is an antilogarithm (antilog)?

An antilogarithm is the inverse of a logarithm. If log_b(x) = y, then x = b^y is the antilog. For example, since log₁₀(100) = 2, the antilog₁₀(2) = 10² = 100. It essentially asks: "what number do I get by raising this base to this power?"

What is the antilog of 2?

It depends on the base. Base 10: antilog(2) = 10² = 100. Base e: antilog(2) = e² ≈ 7.389. Base 2: antilog(2) = 2² = 4. The most common interpretation (in school maths) is base 10, so the antilog of 2 = 100.

How do I calculate antilog manually?

Method 1 — Direct formula: Simply compute b^y. For base 10 and y = 2.5, calculate 10^2.5 = 10² × 10^0.5 = 100 × 3.162 ≈ 316.2.

Method 2 — Log table: Split y into characteristic (integer part) + mantissa (decimal part). Locate the mantissa in a printed antilog table to get a 4-digit number, then position the decimal using the characteristic.

What is the difference between base-10 antilog and natural antilog?

Base-10 antilog = 10^y — the most widely used in school maths, science tables, pH, and the Richter scale.

Natural antilog (exp) = e^y, where e ≈ 2.71828 — used in calculus, finance (continuous compounding), and biological growth models. On a calculator, look for the "e^x" or "exp" key to compute it.

What is the antilog of a negative number?

The antilog of a negative number is always a positive fraction (between 0 and 1). For example, antilog₁₀(−3) = 10⁻³ = 0.001. This is very common in chemistry — a solution with pH 3 has a hydrogen ion concentration of 10⁻³ = 0.001 mol/L.

How is antilog used in real life?

Antilog is used everywhere logarithms are used — just in reverse: Chemistry (pH to H⁺ concentration), Seismology (Richter scale magnitude to energy), Finance (continuous compound growth), Acoustics (decibels to power ratio), and Computer Science (binary data capacity). Anywhere you see a log scale, converting back to a linear value requires an antilog.

What bases can be used for antilog?

Any positive number except 1 can be a valid base. Base 1 is excluded because 1 raised to any power always equals 1, making it useless. The three most common bases are 10, e, and 2. Our calculator supports all common bases and any custom base you enter.

What is the antilog table and how is it used?

An antilog table lists 10^m for mantissa values m from 0.00 to 0.99. To use it: (1) Split your log value y into characteristic (integer part) and mantissa (decimal part). (2) Find the 4-digit antilog number from the table using the first two decimal digits as the row and the third as the column. (3) Adjust the decimal point based on the characteristic: position = characteristic + 1 digits from the left. Try our Advanced tab above for an interactive demonstration.