# The Natural Logarithmic Function: Differentiation and Integration

## Differentiation and integration of natural logarithms is based on the property $\frac{d}{dx}ln(x) = \frac{1}{x}$.

#### Key Points

• The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x).

• The natural logarithm can be integrated using integration by parts: $\int ln(x)dx = xln(x) - x + C$.

• The derivative of the natural logarithm eads to the Taylor series for ln(1 + x) around 0: $ln(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - ... for \left | x \right | \leq 1$unless x = -1.

#### Terms

• of a real number, that cannot be written as the ratio of two integers

• of or relating to a number that is not the root of any polynomial that has positive degree and rational coefficients

#### Figures

1. ##### Interactive Graph: Taylor Polynomial

The Taylor polynomials for  only provide accurate approximations in the range −1 < x ≤ 1. Note that, for x > 1, the Taylor polynomials of higher degree are worse approximations.

The natural logarithm, generally written as ln(x), is the logarithm with the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.

The derivative of the natural logarithm is given by $\frac{d}{dx}ln(x) = \frac{1}{x}$.

This leads to the Taylor series for ln(1 + x) around 0: $ln(1+x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - ... for \left | x \right | \leq 1$, unless $x = -1$.

(Figure 1) is a picture of ln(1 + x) and some of its Taylor polynomials around 0. These approximations converge to the function only in the region −1 < x ≤ 1; outside of this region the higher-degree Taylor polynomials are worse approximations for the function.

Substituting x − 1 for x, we obtain an alternative form for ln(x) itself: $ln(x) = (x - 1) - \frac{(x - 1)^{2}}{2} + \frac{(x - 1)^{3}}{3} - ... for \left | x - 1 \right | \leq 1$ unless $x = 0$.

By using Euler transform, we reach the following equation, which is valid for any x with absolute value greater than 1: $ln\frac{x}{x-1} = \frac{1}{x} + \frac{1}{2x^{2}} + \frac{1}{3x^{3}} + ...$.

The natural logarithm allows simple integration of functions of the form g(x) = f '(x)/f(x): an antiderivative of g(x) is given by ln(|f(x)|). This is the case because of the chain rule and the following fact: $\frac{d}{dx}(ln(\left | x \right |)) = \frac{1}{x}$.

In other words, $\int \frac{1}{x}dx = ln\left | x \right | + C$

and $\int \frac{f'(x)}{f(x)}dx = ln\left | f(x) \right | + C$.

Here is an example in the case of g(x) = tan(x): $\int tan (x)dx = \int \frac{sin (x)}{cos (x)}dx$

$\int tan (x)dx = \int \frac{\frac{-d}{dx}cos (x)}{cos (x)}dx$

Letting f(x) = cos(x) and f'(x)= – sin(x):

$\int tan (x)dx = -ln\left | cos(x) \right | + C$

where C is an arbitrary constant of integration.

The natural logarithm can be integrated using integration by parts: $\int ln(x)dx = xln(x) - x + C$.

#### Key Term Glossary

antiderivative
an indefinite integral
##### Appears in these related concepts:
approximation
An imprecise solution or result that is adequate for a defined purpose.
##### Appears in these related concepts:
chain rule
formula for computing the derivative of the composition of two or more functions
##### Appears in these related concepts:
converge
of a sequence, to have a limit
##### Appears in these related concepts:
derivative
a measure of how a function changes as its input changes
##### Appears in these related concepts:
differentiation
the process of determining the derived function of a function
##### Appears in these related concepts:
e
the base of the natural logarithm, 2.718281828459045…
##### Appears in these related concepts:
function
a relation in which each element of the domain is associated with exactly one element of the co-domain
##### Appears in these related concepts:
integration
the operation of finding the region in the x-y plane bound by the function
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irrational
of a real number, that cannot be written as the ratio of two integers
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logarithm
the exponent by which another fixed value, the base, must be raised to produce that number
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natural logarithm
the logarithm in base e
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polynomial
an expression consisting of a sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a non-negative integer power
##### Appears in these related concepts:
series
the sum of the terms of a sequence
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Taylor series
A representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.
##### Appears in these related concepts:
transcendental
of or relating to a number that is not the root of any polynomial that has positive degree and rational coefficients