Definition

1. Noun:
   • A logarithm to the base e: The natural logarithm of a number is the power to which the mathematical constant e (approximately 2.71828) must be raised to equal that number. It is the inverse function of the exponential function with base e.

Usage

• The natural logarithm is the standard logarithm used in advanced mathematics, calculus, and many scientific fields.
• It is typically denoted as ln(x), where is the argument. For example, ln(7.389...) = 2 because ² ≈ 7.389.
• It is used to solve equations involving exponential growth or decay, such as in models for population growth, radioactive decay, and compound interest.

Examples

• Noun:
  • The derivative of the natural logarithm of x is 1/x.
  • To solve the equation e^y = 10, you take the natural logarithm of both sides.
  • In this formula, the rate constant k is proportional to the natural logarithm of the concentration ratio.

Advanced Usage

• "ln" as a function: The notation ln(x) is universally understood to mean the natural logarithm. For example, "Find ln(1)" yields the answer 0.
• Properties: Key properties include ln(1) = 0, ln(e) = 1, and ln(a*b) = ln(a) + ln(b). These properties are fundamental for simplifying complex expressions.

Variants and Related Words

• Logarithm (n): The general concept. A logarithm answers the question: "To what exponent must we raise a base to get a certain number?" The natural logarithm is a specific type with base .
• Common logarithm (n): A logarithm to the base 10, typically written as log(x).
• Exponential function (n): The function ^x, which is the inverse of the natural logarithm function.

Synonyms

• Napierian logarithm (n): A less common synonym, named after the mathematician John Napier.
• Hyperbolic logarithm (n): An archaic term, now rarely used.

Related Phrases

• Natural log: A common abbreviated form used in speech and informal writing. For example, "Take the natural log of both sides of the equation."
• Log to the base e: A descriptive phrase equivalent to "natural logarithm."

Noun

1. a logarithm to the base e