The natural logarithm is the logarithm having base e, where
This function can be defined
This definition means that e is the unique number with the property that the area of the region bounded by the hyperbola, the x-axis, and the vertical lines and is 1. In other words,
The notation is used in physics and engineering to denote the natural logarithm, while mathematicians commonly use the notation . In this work, denotes a natural logarithm, whereas denotes the common logarithm.
There are a number of notational conventions in common use for indication of a power of a natural logarithm. While some authors use (i.e., using a trigonometric function-like convention), it is also common to write .
Common and natural logarithms can be expressed in terms of each other as
The natural logarithm is especially useful in calculus because its derivative is given by the simple equation
whereas logarithms in other bases have the more complicated derivative
The natural logarithm can be analytically continued to complex numbers as
where is the complex modulus and is the complex argument. The natural logarithm is a multivalued function and hence requires a branch cut in the complex plane, which the Wolfram Language's convention places at .
The principal value of the natural logarithm is implemented in the Wolfram Language as [x], which is equivalent to [, x]. This function is illustrated above in the complex plane.
Note that the inverse trigonometric and inverse hyperbolic functions can be expressed (and, in fact, are commonly defined) in terms of the natural logarithm, as summarized in the table below. Therefore, once these definition are agreed upon, the branch cut structure adopted for the natural logarithm fixes the branch cuts of these functions.
The Mercator series
gives a Taylor series for the natural logarithm.
Continued fraction representations of logarithmic functions include
(Lambert 1770; Lagrange 1776; Olds 1963, p. 138; Wall 1948, p. 342) and
(Euler 1813-1814; Wall 1948, p. 343; Olds 1963, p. 139).
For a complex number, the natural logarithm satisfies
where is the principal value.
Some special values of the natural logarithm include
Natural logarithms can sometimes be written as a sum or difference of "simpler" logarithms, for example
which follows immediately from the identity
Plouffe (2006) found the following beautiful identities:
Euler, L. "Commentatio in fractionem continuam qua illustris La Grange potestates binomiales expressit." Mém. de l'Acad. imperiale des sciences de St. Pétersbourg6, 1813-1814.
Lagrange, J.-L. "Sur l'usage des fractions continues dans le calcul intégral." Nouv. mém. de l'académie royale des sciences et belles-lettres Berlin, 236-264, 1776. Reprinted in Oeuvres, Vol. 4, pp. 301-302.
Lambert, J. L. Beiträge zum Gebrauch der Mathematik und deren Anwendung. Theil 2. Berlin, 1770.
Olds, C. D. Continued Fractions. New York: Random House, 1963.
Plouffe, S. "Identities Inspired from Ramanujan Notebooks (Part 2)." Apr. 2006. http://www.lacim.uqam.ca/~plouffe/inspired2.pdf.
Wall, H. S. Analytic Theory of Continued Fractions. New York: Chelsea, 1948.
Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae, 30th ed. Boca Raton, FL: CRC Press, 1996.Referenced on Wolfram|Alpha: Natural LogarithmCITE THIS AS:
Weisstein, Eric W. "Natural Logarithm." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/NaturalLogarithm.html
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Next: About this document ...
THE INTEGRATION OF EXPONENTIAL FUNCTIONS
The following problems involve the integration of exponential functions. We will assume knowledge of the following well-known differentiation formulas :
where , and
where a is any positive constant not equal to 1 and is the natural (base e) logarithm of a . These formulas lead immediately to the following indefinite integrals :
As you do the following problems, remember these three general rules for integration :
where n is any constant not equal to -1,
where k is any constant, and
Because the integral
where k is any nonzero constant, appears so often in the following set of problems, we will find a formula for it now using u-substitution so that we don't have to do this simple process each time. Begin by letting
du = k dx ,
(1/k)du = dx .
Now substitute into the original problem, replacing all forms of x, and getting
We now have the following variation of formula 1.) :
The following often-forgotten, misused, and unpopular rules for exponents will also be helpful :
Most of the following problems are average. A few are challenging. Knowledge of the method of u-substitution will be required on many of the problems. Make precise use of the differential notation dx and du and always be careful when arithmetically and algebraically simplifying expressions.
- PROBLEM 1 : Integrate .
Click HERE to see a detailed solution to problem 1.
- PROBLEM 2 : Integrate .
Click HERE to see a detailed solution to problem 2.
- PROBLEM 3 : Integrate .
Click HERE to see a detailed solution to problem 3.
- PROBLEM 4 : Integrate .
Click HERE to see a detailed solution to problem 4.
- PROBLEM 5 : Integrate .
Click HERE to see a detailed solution to problem 5.
- PROBLEM 6 : Integrate .
Click HERE to see a detailed solution to problem 6.
- PROBLEM 7 : Integrate .
Click HERE to see a detailed solution to problem 7.
- PROBLEM 8 : Integrate .
Click HERE to see a detailed solution to problem 8.
- PROBLEM 9 : Integrate .
Click HERE to see a detailed solution to problem 9.
- PROBLEM 10 : Integrate .
Click HERE to see a detailed solution to problem 10.
- PROBLEM 11 : Integrate .
Click HERE to see a detailed solution to problem 11.
- PROBLEM 12 : Integrate .
Click HERE to see a detailed solution to problem 12.
Click HERE to return to the original list of various types of calculus problems.
Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :