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Derivative of lnx – a brief overview of its properties and origins

Derivative of lnx – formula

The general formula for the derivative of lnx is (lnx)’ = 1/x. Let us try to find out, how we got this expression. It is derived from the formula of the derivative of loga(x), with the assumption that a = e. As you might know e is the base of natural logarithm, but we will touch this in more details later. So, we got the following: (lnx)’ = (loge(x))’ = 1/(x*ln(e)) = 1/x. Here we used the property of logarithm logaa = 1. If after the natural logarithm sign we got a complicated function, the derivative would be: (lnu)’ = 1/u*u’. Let us solve a couple of examples of finding the derivative of lnx before we jump into some more details considering the properties and origins of a natural logarithm. The first problem is to find the derivative of function y(x) = lnx/3. According to the property of derivative of lnx we get this: y’(x) = 1/3*(lnx)’=1/3*1/x=1/(3*x). So the answer is y’(x) = 1/(3*x). Another example is following: y(x) = ln(3*x). y’(x) = (1/(3*x))*3*x’ = 1/x * 1 = 1/x. The answer is 1/x.

Natural logarithm – definition and properties

Moving on from finding the derivative of lnx, let us define the natural logarithm instead. The vatural logarithm is a logarithm with base e, where e – an irrational constant approximately equal to 2.718281828. The natural logarithm is referred usually to as lnx, loge (x) or sometimes simply log (x), if the base e understood to be there. In other words, the natural logarithm of x – a measure of the power in which you have to raise a number e, in order to obtain x. Let us look at some examples:

  • ln (7,389…) is equal to 2 because e2 = 7.389…
  • ln (e) is 1 because e1 = e
  • ln (1) is 0 because 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 simplicity of this definition is consistent with many other formulas that use the natural logarithm has led to the appearance of the name “natural”. This definition can be extended to the complex numbers, as we will be discussing below. If we consider the natural logarithm of a real function of a real variable, it is the inverse of the exponential function, leading to the equations: e ^ (ln (a)) = a (a> 0) and ln (e ^ a) = a. Like all logarithms, the natural logarithm maps multiplication in addition: ln (xy) = ln (x) + ln (y). In terms of general algebra, logarithmic function makes an isomorphism of positive real numbers group of under multiplication to the group of real numbers under addition: ln: R+ -> R.

The logarithm may be defined for any positive base other than 1, not only e, but for the logarithms of other bases different from those of the natural logarithm only by a constant factor, and is usually defined in terms of the natural logarithm. Logarithms are useful for solving equations in which unknown are present as the exponent. For example, logarithms are used to find the decay constant for the well-known half-life or time spent in dealing with radioactive decay problems. They play an important role in many areas of mathematics and applied sciences, applied in the field of finance for many tasks, including calculation of compound interest.

The history of the natural logarithm and the use of the derivative of lnx

The first mention of the natural logarithm did Nicholas Mercator in Logarithmotechnia, published in 1668, although the mathematics teacher John Spaydell in 1619 created the table of natural logarithms. Previously, it was called the hyperbolic logarithm, as it corresponds to the area of a hyperbole. Sometimes it is called the Napier logarithm, although the original meaning of this term has been somewhat different. The derivative of lnx always refers to the natural logarithm. Symbols lgx and logx depend on the context and traditions, described below. The natural logarithm is denoted by lnx, the logarithm with the base 10 – through lgx, and other bases are specified explicitly with «log» symbol. In many works on discrete mathematics, cybernetics, computer science authors use the notation «log (x)» for the logarithm to the base 2, but this agreement is not universally accepted and requires clarification either in the list of used symbols, or, if such a list is absent, a footnote or comment on the first use. The parentheses around the argument of the logarithm (if it does not lead to an erroneous reading of the formula) are usually omitted, and in the construction of the logarithm of the power indicator is credited directly to the logarithm.

Designation of the natural logarithm as lnx, where x is the argument, was introduced by the American mathematician Irving Stringham in 1893. Mathematicians, statisticians and engineers commonly refer to the natural logarithm as a «log (x)», or «ln (x)», and refer to the logarithm with the base 10 – «log10 (x)». Some engineers, biologists and other specialists will write «ln (x)» (or sometimes «loge (x)»), when they mean the natural logarithm, and write «log (x)» they mean log10 (x). In theoretical computer science, information theory and the cryptography «log (x)» usually means the logarithm with the base 2 «log2 (x)» (though often instead it is written lg (x)). The most commonly used programming languages and software packages, including C, C ++, SAS, MATLAB, FORTRAN and BASIC use function «log» or «LOG» that refers to the natural logarithm. In the hand-held calculators, the natural logarithm is denoted by ln, whereas log is used to designate the logarithm with the base 10.

At first, it may seem that because our number system has a base 10, this base is more “natural” than base e. But mathematically the number 10 is not particularly significant. Its use has more to do with culture, it is common to many number systems, and this is due, probably, to the number of fingers in humans. Some cultures based their numbering system on other bases: 5, 8, 12, 20 and 60. loge is a “natural” logarithm because it occurs automatically and appears in mathematics very often. If logarithm has the base e, the derivative of lnx is a 1 / x, and when x = 1, this derivative of lnx is equal to 1. Another reason why the logarithm with base e is the most natural is that it can be quite simply defined in terms of a simple integral or Taylor series, which is not true for other logarithms. For example, there are some simple series with natural logarithms. Pietro Mengoli and Nicholas Mercator called them logarifmus naturalis few decades before Newton and Leibniz developed differential and integral calculus.

Derivative of lnx: Complex logarithms


The exponential function can be extended to a function which gives a complex number of the form ex for any arbitrary complex numbers x, it uses an infinite series with complex x. This exponential function can be inverted to a form of a complex logarithm, which will have most of the properties of conventional logarithms. There are, however, two difficulties: there is no x, for which ex = 0, and it turns out that e2πi = 1 = e0. Since the multiplicative property is valid for the complex exponential function, ez = ez + 2nπi for all complex z and integers n. The log cannot be defined on the whole complex plane, and even then it is multi-valued – any complex logarithm can be replaced by “equivalent” logarithm by adding any integer multiple of 2pi. The complex logarithm can be unique only on a section of the complex plane. For example, lni = 1/2 πi or 5/2 πi or -3/2 πi, etc., and although i4 = 1, 4 logi can be defined as 2πi, or 10πi or -6 πi, and etc. So in this article we discussed very general information about the derivative of lnx, the natural logarithm in general and its usage in different branches of human activity.

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