Natural Logarithm Calculator

The natural logarithm is a mathematical function used in various fields such as finance, physics, and engineering. It is the logarithm to the base e, where e is a mathematical constant approximately equal to 2.71828.
ln(1)=  0ln(1)=\;00
Photo of calculator author Wes NolteCreated by Wes Nolte.Last updated Apr 25, 2024.

The Natural Logarithm Calculator Formula

Calculating the natural logarithm (ln) involves using a logarithm function with base 'e', where 'e' is the mathematical constant approximately equal to 2.71828. Here is the Natural Logarithm Formula and the steps to calculate the natural logarithm of a number 'x'.

y=ln x=logexy = ln\thinspace x = log_{e}x

  1. Understand the Concept:
    • The natural logarithm of a number 'x', denoted as ln(x), represents the exponent to which the base 'e' (approximately 2.71828) must be raised to obtain the value 'x'. In other words, ln(x) = y means e^y = x.
  2. Use a Scientific Calculator:
    • Most calculators have a natural logarithm function denoted as 'ln' or 'log'. Simply enter the number 'x' and press the 'ln' or 'log' button to get the natural logarithm.
  3. Using Mathematical Formulas:
    • If you want to calculate ln(x) manually without a calculator, you can use the following formula:
    ln⁡(x)=log⁡(x)log⁡(e)ln(x) = rac{log(x)}{log(e)}
    Here, log⁡(x)log(x) represents the base-10 logarithm of 'x', and log⁡(e)log(e) represents the base-10 logarithm of the constant 'e'.
  4. Approximation Methods:
    • There are various approximation methods, such as Taylor series expansion, to estimate ln(x) for specific values of 'x'. These methods are more advanced and are usually implemented in programming or scientific software.

Definition of the Natural Logarithm

The natural logarithm is a mathematical function that describes the logarithmic relationship between a positive real number and the mathematical constant e, which is approximately equal to 2.71828. It is denoted as ln(x), where x is a positive real number. The natural logarithm of a number x is the exponent to which the base e must be raised to produce x. In other words, if y = ln(x), then e to the power of y equals x, or eʸ = x. The natural logarithm has applications in various fields such as mathematics, physics, and finance, and is used to model exponential growth and decay, among other things.
NumberNatural logarithm
10
20.69314718056
31.09861228866
41.38629436111
51.60943791243
102.30258509299
1004.60517018599
10006.90775527898
100009.16290731707
10000011.4472988585

The Exponential Function

The natural logarithm and the exponential function are inverse functions of each other, meaning that the natural logarithm of a number x and the exponential function of a number y cancel each other out. The exponential function is expressed as eʸ, where e is the mathematical constant approximately equal to 2.71828, and y is any real number.The natural logarithm is the inverse of the exponential function, and is expressed as ln(x), where x is any positive real number. If y = ln(x), then eʸ = x, and if y = eˣ, then x = ln(y). This inverse relationship between the natural logarithm and exponential function is useful in various mathematical, scientific, and financial applications.