El logaritmo natural suele ser conocido normalmente como logaritmo neperiano, aunque esencialmente son conceptos distintos. Para más detalles, véase logaritmo neperiano. En matemáticas se denomina logaritmo natural o informalmente logaritmo neperiano al logaritmo cuya base es el número e, un número irracional cuyo valor aproximado es 2,7182818284590452353602874713527. El logaritmo natural se suele denominar como ln(x) o a veces como loge(x) — e incluso en algunos contextos log(x) —, porque para ese número se cumple la propiedad de que el logaritmo vale 1. El logaritmo natural de un número x es entonces el exponente a al que debe ser elevado el número e para obtener x. Por ejemplo, el logaritmo de 7,38905... es 2, ya que e2=7,38905... El logaritmo de e es 1, ya que e1=e. Desde el punto de vista del análisis matemático, puede definirse para cualquier número real positivo x>0 como el área bajo la curva y=1/t entre 1 y x. La sencillez de esta definición es la que justifica la denominación de «natural» para el logaritmo con esta base concreta. Esta definición puede extenderse a los números complejos. El logaritmo natural es entonces una función real con dominio de definición los números reales positivos: y corresponde a la función inversa de la función exponencial:

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The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x) or log(x). This is done in particular when the argument to the logarithm is not a single symbol, 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 e2.0149... = 7.5. The natural log of e itself, ln(e), is 1, because e1 = e, while the natural logarithm of 1, ln(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 (the area being taken as negative when 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 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. The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities: Like all logarithms, the natural logarithm maps multiplication into addition: Thus, the logarithm function is a group isomorphism from positive real numbers under multiplication to the group of real numbers under addition, represented as a function: Logarithms can be defined to 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 are usually defined in terms of the latter. For instance, the binary logarithm is the natural logarithm divided by ln(2), the natural logarithm of 2. 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 the sciences and are used in finance to solve problems involving compound interest. By Lindemann–Weierstrass theorem, the natural logarithm of any positive algebraic number other than 1 is a transcendental number.

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