Natural logarithm

The natural logarithm, written ln x, is the exponent to which the mathematical constant e must be raised to obtain a positive number x.

Exponential function: A function of the form f(x) = aˣ, where the base a is a positive constant other than 1. It reverses the natural logarithm: ln(eˣ) = x.
Logarithm: The inverse operation to exponentiation, giving the exponent needed to produce a specified number. The natural logarithm is the logarithm with the particular base e.
Continuous compounding: A financial model in which interest is compounded continuously, producing growth proportional to e raised to elapsed time. Solving for time in a continuously compounded balance uses a natural logarithm.
Binary logarithm: A logarithm with base 2, written log₂, that measures exponents of powers of two. It is favored in computing and yields information measured in bits.
Euler's number: The irrational constant approximately 2.71828 that is the base of the natural logarithm. Its defining growth properties make it the logarithm's base.
Common logarithm: A logarithm with base 10, commonly written log₁₀ or simply log in some contexts. It uses base 10 instead of e and is common in measurement and engineering.
Exponential growth: Growth in which a quantity's rate of increase is proportional to its current size. Taking ln of an exponentially growing quantity makes its growth linear in time.
Logarithm base 10: The logarithmic function that assigns x the exponent to which 10 must be raised to obtain x. Its conventional notation can differ from ln, especially across disciplines.
Derivative of the natural logarithm: The derivative of ln x is 1/x for positive x. This simple derivative is a central consequence of the logarithm's inverse relationship with eˣ.
Change of base formula: An identity that rewrites a logarithm in one base using logarithms in another base. It gives log_b x = ln x / ln b.

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