The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, log_e x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), log_e(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.

The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln 7.5 is 2.0149..., because e^2.0149... = 7.5. The natural logarithm of e itself, ln e, is 1, because e¹ = e, while the natural logarithm of 1 is 0, since e⁰ = 1.

The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a (with the area being negative when 0 < a < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see Complex logarithm for more.

The natural logarithm function, if considered as a real-valued function of a positive real variable, is the inverse function of the exponential function, leading to the identities:

${\displaystyle {\begin{aligned}e^{\ln x}&=x\qquad {\text{ if }}x{\text{ is strictly positive,}}\\\ln e^{x}&=x\qquad {\text{ if }}x{\text{ is any real number.}}\end{aligned}}}$

Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:

${\displaystyle \ln(x\cdot y)=\ln x+\ln y~.}$

Logarithms can be defined for any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter, ${\displaystyle \log _{b}x=\ln x/\ln b=\ln x\cdot \log _{b}e}$.

Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and scientific disciplines, and are used to solve problems involving compound interest.