Logs, the number line...........and transcendental numbers.............what is Johnny D's ships log?  Capt. Sparrow.......................

Graph of the natural logarithm function. The function slowly grows to positive infinity as x increases and slowly goes to negative infinity as x approaches 0 ("slowly" as compared to any power law of x); the y-axis is an asymptote.

The natural logarithm of a number is its logarithm to the base e, where e is an irrational and transcendental constant 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, simplylog x.^[1] Parentheses are sometimes added for clarity, giving ln(x), log_e(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..., becausee^2.0149... = 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: