Logarithm

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The logarithm is a strictly concave (increasing) monotonic function comprised in the set of positive real numbers and is the inverse of the exponential function.

In other words, the logarithm is a function that depends on a base and an argument that grows at a decreasing rate of growth.

Recommended articles: natural logarithm, logarithms in econometrics and real numbers.

Logarithm formula

Logarithm formula.

The logarithm expression is made up of a base and an argument.

In this case, the base is x and the argument is z from which we will obtain the logarithm.

But… Of the elements of the previous equation, what is the logarithm?

Most of us tend to think that the logarithm of the previous expression is just logx, but it is not true. The correct answer is logxz since we also need the variable z to be able to calculate the logarithm.

Domain

Given a numerical variable z comprised within the set of real numbers, it is subject to the restriction of adopting only positive reals.

Mastery of the argument.

In other words, the logarithm arguments will only take real numbers strictly (>) greater than zero.

Given a number x comprised within the set of real numbers, it is subject to the restriction of adopting only positive reals greater than 1.

Base domain.

In other words, the bases of logarithms will only take real numbers strictly (>) greater than one.

The most used bases are 2, 10 and e.

The logarithm to base 10 is called the common or decimal logarithm.

The logarithm to base 2 is known as the binary logarithm.

If the base of the logarithm is the number e, then the logarithm is called the natural or natural logarithm.

Representation

Base 10 logarithm of each positive real number on the horizontal axis.

What do we need to calculate the logarithm of a number?

To calculate the logarithm we need two numbers that belong to the set of positive reals and also that one of them is different from one. One number will act as the argument and the other as the base, respectively.

Outcome

Although there are restrictions on the numbers that can be used for the base and the argument, the codomain of the logarithmic function is all real numbers. In other words, we can obtain negative, neutral or positive logarithms since they can take any value of the real line:

Set of the real numbers.

It is important not to confuse the domain of the argument with the domain of the result (codomain).

Examples

App

In finance, logarithms are used to obtain the continuous returns of an asset or financial product.

In economics, both in microeconomics and macroeconomics, they are used to express the aversion to risk of economic agents in utility functions. They are also used to do monotonous transformations of utility functions.

In econometrics, the scale of the variables is transformed to facilitate their interpretation.

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