natural logarithm
Học thuậtThân thiện
Definition
- 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
- a logarithm to the base e