Log Base 10 Calculators
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What Is Log Base 10?
The common logarithm (log₁₀) of a number x is the exponent y such that 10^y = x:
log₁₀(x) = y means 10^y = x
Examples: log₁₀(1) = 0; log₁₀(10) = 1; log₁₀(100) = 2; log₁₀(1000) = 3; log₁₀(0.001) = −3.
The common log is the inverse of the exponential function 10^x.
Key Properties of Log₁₀
- log(a × b) = log(a) + log(b)
- log(a / b) = log(a) − log(b)
- log(a^n) = n × log(a)
- log(1) = 0; log(10) = 1
- log of a negative number is undefined in real numbers
- log(0) is undefined (approaches −∞)
Calculating Log₁₀
On a scientific calculator, the LOG button computes log₁₀. Conversely, 10^x (the antilog) converts back. To convert between log₁₀ and natural log (ln):
ln(x) = 2.303 × log₁₀(x)
log₁₀(x) = ln(x) / 2.303
Log Base 10 in Science
pH
pH = −log₁₀[H⁺]. A change of 1 pH unit = a 10-fold change in H⁺ concentration. This is why the pH scale compresses the enormous range of hydrogen ion concentrations (from 1 M to 10⁻¹⁴ M) into a 0–14 scale.
Decibels (dB)
Sound intensity in decibels: dB = 10 × log₁₀(I/I₀), where I₀ = 10⁻¹² W/m² (threshold of hearing). A 10 dB increase = 10× greater intensity; 20 dB = 100×; 60 dB = 10⁶×.
Richter Scale
The Richter magnitude scale uses log₁₀ of ground motion amplitude. A magnitude 7 earthquake has 10× more ground motion amplitude than magnitude 6, and releases ~31.6× more energy.
Dilution Series in Biology
Serial dilutions are typically 10-fold — log₁₀ transforms convert dilution factors to a linear scale. Viral titers, bacterial counts, and antibody concentrations are routinely plotted on log₁₀ axes.
Glossary
Frequently Asked Questions
Log base 10 (log₁₀ or common log) is the power to which 10 must be raised to equal a given number: log₁₀(x) = y means 10^y = x. For example, log₁₀(1000) = 3 because 10³ = 1000; log₁₀(0.01) = −2 because 10⁻² = 0.01. It compresses large ranges of values into a manageable scale by converting multiplication into addition.
log (log₁₀) is the common logarithm with base 10 — used in pH, decibels, and engineering. ln (natural log) uses base e ≈ 2.71828 — used in calculus, exponential growth/decay, and thermodynamics. Conversion: ln(x) = 2.303 × log₁₀(x). In pure mathematics, 'log' often means natural log by convention, but in applied sciences it typically means log₁₀.
pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in mol/L. For [H⁺] = 10⁻⁷ mol/L: pH = −(−7) = 7 (neutral). For [H⁺] = 10⁻³ mol/L: pH = 3 (acidic). Each unit change in pH = a 10-fold change in H⁺ concentration — the log scale makes the enormous range of physiologically relevant H⁺ concentrations fit neatly in a 0–14 range.
Logarithmic scales are useful when data spans many orders of magnitude. Concentration ranges in biology often vary 10⁶-fold or more (pH, antibody titers, gene expression). Plotting these on a linear axis compresses most data into a tiny range. A log scale spreads data evenly and makes fold changes visually intuitive — each equal interval represents the same proportional change, not the same absolute change.