About

Computing log_b(x) when your calculator only provides ln or log₁₀ requires the change of base formula. An incorrect base conversion propagates through every dependent calculation - entropy estimates in information theory shift, pH computations in chemistry become unreliable, and signal decibel measurements lose meaning. This tool computes the exact logarithmic value using log_b(x) = ln(x) ÷ ln(b) and displays the full intermediate steps so you can verify each stage. It handles fractional and irrational bases, flags undefined cases (x ≤ 0, b ≤ 0, b = 1), and rounds to a user-specified precision up to 15 significant digits.

Limitation: results rely on IEEE 754 double-precision floating point, which caps reliable precision at roughly 15 - 17 significant decimal digits. For arguments exceeding 10³⁰⁸ or below 10⁻³⁰⁸, overflow or underflow may occur. The tool assumes real-valued logarithms only; complex results for negative arguments are not supported.

Formulas

The change of base formula expresses a logarithm in base b using any other base a:

log_b(x) = log_a(x)log_a(b)

When the intermediary base a is chosen as e (Euler's number), this simplifies to the natural logarithm form used internally by this calculator:

log_b(x) = ln(x)ln(b)

Equivalently, using the common logarithm (base 10):

log_b(x) = log₁₀(x)log₁₀(b)

The reciprocal identity follows directly:

log_b(a) = 1log_a(b)

Where: x = the argument (must be > 0). b = the target base (must be > 0 and ≠ 1). a = any convenient intermediary base (typically e or 10). ln = natural logarithm (log_e). Domain constraint: x ∈ (0, ∞), b ∈ (0, 1) ∪ (1, ∞).

Reference Data

Base (b)	Common Name	Symbol	Primary Domain	log_b(2)	log_b(10)	log_b(100)
2	Binary Logarithm	lb / log₂	Computer Science, Information Theory	1.0000	3.3219	6.6439
e ≈ 2.7183	Natural Logarithm	ln	Calculus, Physics, Differential Equations	0.6931	2.3026	4.6052
3	Ternary Logarithm	log₃	Ternary Computing, Number Theory	0.6309	2.0959	4.1918
4	Quaternary Logarithm	log₄	DNA Encoding (4 nucleotides)	0.5000	1.6610	3.3219
5	Quinary Logarithm	log₅	Tally Systems	0.4307	1.4307	2.8614
8	Octal Logarithm	log₈	Unix Permissions, Legacy Computing	0.3333	1.1073	2.2146
10	Common Logarithm	log / log₁₀	Engineering, Chemistry (pH), Acoustics (dB)	0.3010	1.0000	2.0000
12	Duodecimal Logarithm	log₁₂	Dozenal Society, Time (12 hours)	0.2789	0.9266	1.8532
16	Hexadecimal Logarithm	log₁₆	Memory Addressing, Color Codes	0.2500	0.8305	1.6610
20	Vigesimal Logarithm	log₂₀	Maya/Aztec Numeral Systems	0.2311	0.7686	1.5372
60	Sexagesimal Logarithm	log₆₀	Babylonian Math, Time/Angles	0.1693	0.5627	1.1255
64	Base-64 Logarithm	log₆₄	Base64 Encoding	0.1667	0.5537	1.1073
100	Centesimal Logarithm	log₁₀₀	Percentage Scaling	0.1505	0.5000	1.0000
256	Byte Logarithm	log₂₅₆	Byte-level Data (8-bit)	0.1250	0.4152	0.8305
1024	Kibi Logarithm	log₁₀₂₄	Storage Units (KiB, MiB, GiB)	0.1000	0.3322	0.6644

Frequently Asked Questions

When b = 1, ln(1) = 0, placing zero in the denominator of the formula ln(x) ÷ ln(b). Division by zero is undefined. Conceptually, 1 raised to any power always equals 1, so no exponent can produce a value other than 1. The logarithm base 1 does not exist in real analysis.

IEEE 754 double-precision provides approximately 15 to 17 significant decimal digits. For arguments near 10³⁰⁸ (the upper limit), Math.log still returns valid results because logarithms compress magnitude. However, for arguments extremely close to 1 (e.g., 1.0000000000000002), cancellation error in ln(x) can reduce accuracy to fewer than 10 reliable digits. This calculator displays up to 15 decimal places; treat the last 2 - 3 digits as uncertain in edge cases.

Yes. The formula log_b(x) = ln(x) ÷ ln(b) works for any valid base, including π, e², or 0.5. Bases between 0 and 1 yield negative denominators, which means the resulting logarithm has the opposite sign compared to bases greater than 1. This is mathematically correct: log_0.5(8) = −3 because 0.5⁻³ = 8.

The conversion factor is log₂(10) ≈ 3.3219. To convert: log₂(x) ≈ 3.322 × log₁₀(x). For example, a number with 3 decimal digits (order 10³) requires roughly 3.322 × 3 ≈ 10 binary bits. This is why 1024 (2¹⁰) approximates 1000.

No. The intermediary base cancels algebraically. Whether you compute ln(x) ÷ ln(b) or log₁₀(x) ÷ log₁₀(b), the quotient is identical. This calculator shows both pathways in the step-by-step breakdown so you can verify numerically. Minor floating-point discrepancies (on the order of 10⁻¹⁵) may appear due to rounding in the last bits.

Shannon entropy is defined as H = −n∑i=1 p_i log_b(p_i). Choosing base 2 gives entropy in bits; base e gives nats; base 10 gives hartleys. Converting between these units is a direct application of the change of base formula: multiply by log_b₂(b₁). For example, 1 nat = log₂(e) ≈ 1.4427 bits.