 |
Business | Industries | Finance | Tax |
| Index: > A B C D E F G H I J K L M N O P Q R S T U V W X Y Z |
|
|||||
| First Prev [ 1 2 ] Next Last |
ln(x)
Mathematicians generally understand either "ln(x)" or "log(x)" to mean loge(x), i.e., the natural logarithm of x, and write "log10(x)" if the base-10 logarithm of x is intended. Engineers, biologists, and some others write only "ln(x)" or (occasionally) "loge(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log10(x). Sometimes also Log(x) (capital L) is used to mean log10(x), by those people who use log(x) with a lowercase l to mean loge(x).
Most of the reason for thinking about base-10 logarithms became obsolete shortly after about 1970 when hand-held calculators became widespread (for more on this point, see common logarithm). Nonetheless, since calculators are made and often used by engineers, the conventions to which engineers were accustomed continued to be used on calculators, so now most non-mathematicians take "log(x)" to mean the base-10 logarithm of x and use only "ln(x)" to refer to the natural logarithm of x. As recently as 1984, Paul Halmos in his autobiography heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. (The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley.) At the time of this writing (2004), some mathematicians have adopted the "ln" notation, but most use "log".
To avoid all confusion, Wikipedia uses the notation ln(x) for the natural logarithm of x and log10(x) for the base-10 logarithm of x.
This function is the inverse function of the exponential function, thus it holds
In other words, the logarithm function is a bijection from the set of positive real numbers to the set of all real numbers. More precisely it is an isomorphism from the group of positive real numbers under multiplication to the group of real numbers under addition.
Logarithms can be defined to any positive base other than 1, not just e, and they are always useful for solving equations in which the unknown appears as the exponent of some other quantity.
Initially, it seems that in a world using base 10 for nearly all calculations, this base would be more "natural" than base e. The reason we call ln(x) "natural" is twofold: first, the natural logarithm can be defined quite easily using a simple integral or Taylor seriesIn mathematics, the Taylor series of an infinitely often differentiable real (or complex) function f defined on an open interval a − r a + r is the power series : Here, n is the factorial of n and f n a denotes the n''th derivative of f at the point as will be explained below; this is not true of other logarithms. Second, expressions in which the unknown variable appears as the exponent of e occur much more often than exponents of 10 (because of the "natural" properties of the exponential function which allow it to describe growth and decay behaviors), and so the natural logarithm is more useful in practice. To put it concretely, consider the problem of differentiating a logarithmic function:
When x is equal to 1, and the base (b) is e, then the slope of the graph will be 1.
Formally, ln(a) may be defined as the area under the graph ( integralThis article deals with the concept of an integral in mathematical calculus. For other meanings of "integral" see integration. In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. Unlike the process of differe) of 1/x from 1 to a, that is,
This defines a logarithm because it satisfies the fundamental property of a logarithm:
This can be shown by defining and using the substitution rule of integrationIn calculus, the substitution rule is an important tool for finding antiderivatives and integrals. It is the counterpart to the chain rule of differentiation. Suppose f ''x is an integrable function, and φ t is a continuously differentiable function w as follows:
The number e can then be defined as the unique real number a such that .
Alternatively, if the exponential function has been defined first using an infinite series, the natural logarithm may be defined as its inverse function, meaning ln(x) is that number for which Since the range of the exponential function is all positive real numbers and since the exponential function is strictly increasing, this is well-defined for all positive x.
| Politics | Business | Finance |
 |
 |
 |
|