Logarithm Calculator

Logarithm, Change-of-Base & Scales Calculator

Evaluate natural, common, binary, and arbitrary-base logarithms, verify algebraic log laws, and model physical decibel/pH scales

Standard Logarithm Presets

Solve Target

Argument (x) Must be positive real (x > 0)

Base (b)

b > 0 and b ≠ 1

Exponent / Power (y) Real power exponent

Evaluated Logarithm: log_b(x)

2.000000

Exact Integer Result

log₁₀(100.0) = 2.000000 • Exponential Verification: 10² = 100

Natural Logarithm (ln x) 4.605170 Base-e (Euler’s number)

Common Logarithm (log₁₀ x) 2.000000 Order of magnitude (Base 10)

Binary Logarithm (log₂ x) 6.643856 Information entropy (bits)

Change-of-Base Quotient ln(x) / ln(b) 4.6052 / 2.3026

Parameters for Logarithmic Identity Verification

Base (b)

Factor u

Factor v

Power k

Product Rule: log_b(u · v) = log_b(u) + log_b(v)

2.000000 = 2.000000

log₁₀(100) = log₁₀(25) + log₁₀(4) = 1.3979 + 0.6021

Quotient Rule: log_b(u / v) 0.795880 log_b(u) – log_b(v)

Power Rule: log_b(u^k) 4.193820 k · log_b(u)

Reciprocal Identity: log_b(1 / u) -1.397940 -log_b(u)

Base Inversion: log_(1/b)(u) -1.397940 -log_b(u)

Logarithmic Application Scale

Measured Power (P) Watts (W)

Reference Baseline (P₀) Baseline datum (e.g. 1W, 1mW, 20µPa)

Calculated Logarithmic Scale Value

+20.00 dB

Power Gain: 100.00× Reference Datum

Linear Dimension Ratio 100.00× P / P₀

Orders of Magnitude 2.000 Decades Decadic exponent factor

Half / Double Interval +3.01 dB per 2× Binary doubling step

Physical Classification Power Gain (+20 dB) Acoustic / RF signal metric