What is the Natural Logarithm?
The natural logarithm is a type of logarithm that uses the constant e (approximately 2.718) as its base. It is written as ln(x), where x is a positive real number. It answers the question: To what power must you raise e to get x? For example, ln(7.39) ≈ 2, because e² ≈ 7.39. In other words, if ln(x) = y, then eʸ = x.
The constant e that the natural logarithm uses was first discovered in the context of problems involving continuous growth, particularly in relation to calculating the area under a hyperbola. It was formalized by mathematicians such as Jacob Bernoulli and later Leonhard Euler. Bernoulli encountered the number while studying compound interest, where it appears as a limiting value, but e is more fundamentally tied to exponential growth and calculus. It can also be defined as the limit of (1 + 1/n)ⁿ as n approaches infinity, or as the sum of the infinite series 1 + 1/1! + 1/2! + 1/3! + …
Natural logarithms are widely used in mathematics, science, and statistics, especially when modeling processes that involve exponential growth or exponential decay. For example, natural logs appear in formulas for compound interest, population growth, radioactive decay, and many statistical models, including logistic regression and exponential smoothing.
Properties
Some key properties of the natural logarithm includes:
- ln(1) = 0 because e⁰ = 1
- ln(e) = 1 because e¹ = e
- ln(a × b) = ln(a) + ln(b)
- ln(a ÷ b) = ln(a) – ln(b)
- ln(aʳ) = r × ln(a)
In calculus and advanced modeling, natural logarithms are especially useful because the derivative of ln(x) is 1/x, and the function grows slowly compared to linear or exponential functions.
Natural logarithms are sometimes referred to as logarithms to base e or log base e. They are distinct from common logarithms, which use base 10 and are written as log(x).
« Back to Glossary Index