Natural logarithm
![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 (](/Images/godic/202501/29/Log.svg0945.png")
![](/Images/godic/202501/29/Euler's_formula.svg0945.png")
![The hyperbola y = 1/x (red curve) and the area from x = 1 to 6 (shaded in orange). This area equals the natural logarithm of 6.](/Images/godic/202501/29/1_over_x_integral.svg0945.png")
![ln(a) illustrated as the area under the curve f(x) = 1/x from 1 to a. If a is less than 1, the area from a to 1 is counted as negative.](/Images/godic/202501/29/Log-pole-x_1.svg0945.png")
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental 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.