About

An antilogarithm reverses a logarithm. If log_b(y) = x, then the antilog is y = b^x. Errors in antilog computation cascade exponentially. A misplaced decimal in x produces order-of-magnitude errors in the result because exponentiation amplifies input deviations. This tool computes b^x for arbitrary positive bases b ≠ 1, with precision from 0 to 15 decimal places. It handles the three standard bases used in science and engineering: 10 (common logarithm), e (natural logarithm), and 2 (binary logarithm).

Note: this calculator approximates results using IEEE 754 double-precision floating-point arithmetic. Precision degrades for exponents beyond ±308 (base 10) where results overflow to Infinity or underflow to 0. For base 1, the logarithm is undefined. For negative or zero bases, real-valued exponentiation is not defined for all exponents.

Formulas

The antilogarithm is the inverse operation of a logarithm. Given a logarithmic value x and a base b, the antilogarithm recovers the original number:

y = antilog_b(x) = b^x

For the three standard bases:

antilog₁₀(x) = 10^x

antilog_e(x) = e^x = exp(x)

antilog₂(x) = 2^x

The relationship between logarithm and antilogarithm is bidirectional:

log_b(b^x) = x and b^log_b(x) = x

Variable definitions: b = base of the logarithm (b > 0, b ≠ 1). x = exponent (the logarithmic value). y = the antilogarithm result. e ≈ 2.71828182845 (Euler's number).

For change-of-base conversions, the antilogarithm in one base can be expressed via another:

b^x = 10^{x ⋅ log₁₀(b)}

Reference Data

Base (b)	Name	Exponent (x)	Antilog (b^x)	Common Use
10	Common	0	1	pH scale, decibels
10	Common	1	10	Order of magnitude
10	Common	2	100	Scientific notation
10	Common	3	1,000	Thousands scale
10	Common	−1	0.1	Negative mantissa
10	Common	−3	0.001	Milliunit conversion
e ≈ 2.71828	Natural	0	1	Exponential growth
e	Natural	1	2.71828	Euler's number
e	Natural	2	7.38906	Compound interest
e	Natural	−1	0.36788	Decay constant
2	Binary	0	1	Digital logic
2	Binary	8	256	Byte range
2	Binary	10	1,024	Kilobyte (KiB)
2	Binary	16	65,536	16-bit address space
2	Binary	20	1,048,576	Megabyte (MiB)
2	Binary	32	4,294,967,296	32-bit integer max
8	Octal	3	512	Unix permissions
16	Hexadecimal	2	256	Color channels (RGB)
10	Common	6.02	1.047 × 10⁶	Avogadro-scale
10	Common	−7	1 × 10⁻⁷	Neutral pH [H⁺]

Frequently Asked Questions

The logarithm base 1 is undefined because 1^x = 1 for all x. No unique exponent maps to a given value. This calculator rejects base 1 and displays an error.

IEEE 754 double-precision floating point supports values up to approximately 1.7976931348623157 × 10³⁰⁸. For base 10, any exponent x > 308 overflows. For base 2, overflow occurs around x ≈ 1024. The calculator warns when the result exceeds representable range.

pH = −log₁₀([H⁺]). To recover the hydrogen ion concentration, compute [H⁺] = 10^−pH. For example, pH 4 means [H⁺] = 10⁻⁴ = 0.0001 mol/L.

Yes. A negative exponent produces a fractional result. antilog₁₀(−2) = 10⁻² = 0.01. This is standard exponentiation. Very large negative exponents underflow to 0 in floating-point representation.

Match precision to your measurement's significant figures. If your logarithmic value has 4 significant digits, set precision to 4 - 6 decimals. For engineering estimates, 4 decimals is typical. For analytical chemistry or physics constants, use 10 - 15.

They are mathematically identical. antilog_b(x) is simply the notation b^x viewed from the perspective of inverting a logarithm. The term "antilog" provides context: you started with a log value and are recovering the original number.