The concept of function it has multiple uses. If we focus on math, a function is a relationship that exists between two sets, whereby each element of the initial set is assigned a single element of the final set (or none at all). Logarithmic, for its part, is that which is linked to a logarithm: the exponent to which a certain amount needs to be raised to obtain a certain number as a result.
From these ideas, we can advance in the definition of logarithmic function. This is the function whose generic expression can be seen in the image.
In these functions, to is the base, which has to be positive and different from 1. The official way to read this equation is as follows: “the function of x is equal to the logarithm base a of x.” It should be mentioned that it could also be expressed without the use of the expression f (x), but with a variable such as Y, since in this way we could reflect more clearly that the result is a different element, from another set.
It is important to mention that the logarithmic function is the inverse function of the exponential function: the one that is represented by the equation f (x) = aˣ
Among the main characteristics of a logarithmic function, we can mention that its domain (your starting or initial set) are the real numbers positive. It is a continuous function, whose route is R (The images obtained from applying the function correspond to any of the elements of the set formed by the real numbers).
Another property is that the logarithmic function of the base is equal to 1 in all cases. Logarithmic functions, on the other hand, can be increasing or decreasing, as well as convex or concave, depending on the value from the base. To know if they are increasing, just observe if to is greater than 1; on the other hand, if it is greater than 0 and less than 1, then it is decreasing.
Continuing with the properties of the logarithmic function, we can say that in the graph we always find the following two points: (1, 0) and (a, 1), understanding these pairs as values on the axes X and Y, that is, the horizontal and the vertical, respectively. The logarithmic function is also considered injective.
In the field of mathematics, it is known by the name of injective function to the one in which each element of the codomain corresponds only to one of the domain. In other words, in a function of this type, to which the logarithmic also belongs, it cannot be the case that more than one element of the first set has the same image.
When graphing a logarithmic function we obtain a symmetric result to that of the exponential function if we take into account the bisector of the first and third quadrants. By bisector is understood the ray that arises at the vertex of an angle and cuts it into two identical parts. The reason for this phenomenon is that both are inverse or reciprocal of each other.
The logarithmic functions, in short, are those in whose equation the variable is the base or argument of a logarithm. To solve these equations, it is generally a matter of converting the logarithmic equation into one that is equivalent but lacks a logarithm.
In the cases that can be represented with the equation present in the first image, the conversion sets the base of the logarithm as that of a power raised to the x and equate this term to Y. For example, if we have a function of x in which the base is 2, for each element of the codomain we will have to find what number is equal to it if we square it.