Definition

1. Noun:
   • A logarithm to the base 10: A common logarithm is a logarithm whose base is the number 10. It answers the question: "To what power must 10 be raised to obtain a given number?" It is a specific type of logarithm widely used in science and engineering, especially before the advent of calculators.

Usage

• The common logarithm is used to simplify calculations involving multiplication, division, and exponentiation by converting them into addition and subtraction.
• It is often denoted simply as without explicitly writing the base 10, especially in contexts where it is the standard (e.g., many scientific calculators have a button for base 10).
• In mathematical notation, it is written as log₁₀(x) or simply log(x).

Examples

• Noun:
  • The common logarithm of 100 is 2, because 10 raised to the power of 2 equals 100.
  • Before computers, engineers relied on slide rules, which used scales based on the common logarithm.
  • In chemistry, pH is calculated using the common logarithm of the hydrogen ion concentration.

Advanced Usage

• "Characteristic and Mantissa": When dealing with common logarithms of numbers not exact powers of 10, the result is often expressed with a characteristic (the integer part) and a mantissa (the decimal part). For example, log(150) = log(1.5 × 10²) = 2 + log(1.5).
• Scientific Notation: Common logarithms are intrinsically linked to scientific notation. The logarithm of a number in the form is .

Variants and Related Words

• Logarithm (n): The general mathematical concept of which the common logarithm is a specific type. A logarithm is the exponent to which a base must be raised to produce a given number.
• Natural Logarithm (n): A logarithm to the base (approximately 2.71828), denoted as .
• Logarithmic Scale (n): A scale of measurement where values are spaced according to the logarithm of the value, not the value itself. The Richter scale for earthquakes and the decibel scale for sound are based on common logarithms.

Synonyms

• Base-10 logarithm: A direct synonym that explicitly states the base.
• Briggsian logarithm: A historical term named after Henry Briggs, who popularized base-10 logarithms.

Related Phrases

• "Take the log": A common instructional phrase meaning to apply the common logarithm function to a number.
  • To solve this, take the log of both sides of the equation.
• "Log table": A reference table listing the common logarithms of numbers, used for manual calculation.
  • She looked up the value in a log table.

Noun

1. a logarithm to the base 10