The Common Logarithm
Simple Explanation
The common logarithm is a logarithm with base 10. It is written log(x) or log₁₀(x) — when no base is written at all, base 10 is always assumed.
Why Do We Need It?
Base 10 matches our everyday decimal number system, which is why the common logarithm is the default on calculators and shows up throughout science in scales like pH and the Richter scale.
Worked Example
Evaluate a common logarithm
Evaluate log(1000).
Why Does This Work?
By the definition of a logarithm, log₁₀(x) is exactly the exponent that 10 must be raised to in order to reach x — for powers of 10, that exponent is simply the count of zeros after the leading 1.
Real-Life Example
The pH scale
pH is defined as pH = −log[H⁺], the negative common logarithm of hydrogen-ion concentration.
Because it uses a base-10 logarithm, each whole-number change in pH represents a tenfold change in acidity — a direct real-world use of the common logarithm.
Practice
Evaluate log(100,000).
EasyCommon mistake
Assuming an unwritten logarithm base means base e — a bare "log" with no base always means base 10 (common logarithm); base e is always written explicitly as "ln".
Quick Review
- log(x) with no visible base always means log₁₀(x).
- For a power of 10, log gives back the exponent directly.
- The common logarithm underlies scales like pH and the Richter scale.