natural logarithm
LowTechnical / Academic
Definition
Meaning
A logarithm to the base of the mathematical constant e (approximately 2.71828).
The inverse function of the natural exponential function e^x; defined as ln(x) = ∫ from 1 to x of (1/t) dt for x>0. It is the unique logarithm whose derivative is 1/x. It is the logarithm most naturally arising in calculus, pure mathematics, and many scientific models.
Linguistics
Semantic Notes
Almost exclusively used in mathematics, science, engineering, and quantitative fields. It is a specific, well-defined mathematical concept, not a descriptive or metaphorical term. It is distinct from the 'common logarithm' (base 10).
Dialectal Variation
British vs American Usage
Differences
No lexical or definitional differences. Differences are purely orthographic (e.g., 'logarithmise' vs. 'logarithmize' in derivative terms, but these are rare). The notation 'ln' is universal.
Connotations
Identical technical connotations.
Frequency
Frequency is dictated entirely by technical context; no regional variation in usage frequency of the core term.
Vocabulary
Collocations
Grammar
Valency Patterns
the natural logarithm of [NP]ln([NP])[NP] = ln([NP])Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “None”
Usage
Context Usage
Business
Extremely rare, except possibly in highly quantitative finance or economic modelling.
Academic
Standard and frequent in mathematics, physics, engineering, chemistry, biology (e.g., population models), and economics texts.
Everyday
Virtually never used.
Technical
The primary context of use; fundamental in calculus, differential equations, and scientific computing.
Examples
By Part of Speech
verb
British English
- First, logarithmise the equation to isolate the exponent.
- We need to take the natural logarithm of both sides.
American English
- First, logarithmize the equation to isolate the exponent.
- We need to take the natural log of both sides.
adverb
British English
- The values increased logarithmically, with a natural base.
- The data was transformed naturally using ln.
American English
- The values increased logarithmically, with a natural base.
- The data was transformed naturally using ln.
adjective
British English
- The natural logarithmic scale revealed the linear relationship.
- This is a natural logarithm function.
American English
- The natural log scale revealed the linear relationship.
- This is a natural log function.
Examples
By CEFR Level
- The natural logarithm, written as 'ln', is a key concept in advanced maths.
- To solve e^x = 10, you take the natural logarithm of both sides.
- The derivative of the natural logarithm of x is 1/x, a fundamental result in calculus.
- The data exhibited exponential decay, so we applied a natural log transformation to linearise it for regression analysis.
Learning
Memory Aids
Mnemonic
Remember 'n' for natural or 'n' in 'ln'. Think of 'natural' as relating to 'e' (Euler's number), which arises naturally from growth processes.
Conceptual Metaphor
Undoing continuous growth: The natural logarithm 'undoes' or 'unpacks' the effect of continuous compounding or natural growth (modelled by e^x).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- The word 'natural' here is not an adjective for 'nature' (природа) but a technical term meaning 'to the base e'. It is калька with 'натуральный логарифм'.
- In Russian mathematical notation, it is also 'ln', identical to English.
Common Mistakes
- Confusing ln(x) with log₁₀(x) or log₂(x).
- Forgetting that ln(x) is only defined for x > 0.
- Writing 'log(x)' to mean ln(x) in general contexts; in many programming languages 'log()' is ln(), but in many school contexts 'log()' is base 10.
Practice
Quiz
What is the base of the natural logarithm?
FAQ
Frequently Asked Questions
Because its base, e, arises naturally in calculus and models of continuous growth/decay, unlike base 10 which is 'artificial' (based on human counting).
In many scientific and mathematical contexts, 'ln' is specifically log base e, while 'log' (without a base written) can mean log base 10 or, in advanced mathematics/computing, often also means log base e. Always check the context.
Essentially yes, though historically John Napier's original logarithms were not exactly to base e, the term 'Napierian logarithm' is often used synonymously with 'natural logarithm'.
In calculating compound interest continuously, in models of population growth or radioactive decay, in the Richter scale for earthquakes, in information theory (entropy), and in solving time constants in physics/engineering.