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Logarithms


Remember that algebraically the logarithm is an exponent. More precisely, if B > 0 and b 1, then for positive values ​​of x O logarithm in base b of x is denoted by

and is defined as that exponent to which B must be raised to produce x. For example,

Historically, the first logarithms to study were the base 10 called logarithms common. For such logarithms, it is usual to suppress explicit reference to the base and write log x and not . More recently, base two logarithms have played an important role in computational science as they naturally arise in binary numerical systems.

However, the most widely used logarithms in applications are natural logarithms, which have a natural base denoted by the letter and in honor of the Swiss mathematician Leonard Euler, who first suggested its application to the logarithms in the unpublished article written in 1728. This constant, whose value is six decimal places, is

and 2,718282

arises as a horizontal asymptote to the graphic of the equation

y =

The values ​​ofapproach and

x
122,000000
101,1 2,593742
1001,01 2,704814
10001,001 2,716924
10.0001,0001 2,718146
100.0001,00001 2,718268
1.000.0001,000001 2,718280

The fact that y = e, When x and when x is expressed by the limits

and

The exponential function f (x) = is called natural exponential function. To simplify typography, this function is sometimes written as exp x. So, for example, you can see the relationship expressed as

exp (+) = exp () exp ()

This notation is also used by computational resources, and it is typical to access the function with some variation of the EXP command.

Next: Logarithmic Functions