The term comes from the Greek roots “logos” which refers to proportion and “arithmos”, which means number, in fact, logarithmic scales are based precisely on numerical proportions. In a broad sense, they are inverse mathematical operations when raising a number to the power. **Logarithms are** used to solve equations that can be presented to us in any area where we develop either consciously or unconsciously.

A simple example would be if we raise 2 ^{4} the result is 16, in this case the inverse or logarithmic operation would be Log _{2} (16) = 4, it starts from the fact that the potentiation (16) and the base (2) is obtained the exponent (4). It is named by the following formula or equation: log _{b} (X) = y yes and only if X = b ^{and} y is read: the logarithm base b of the number x is y.

Index

## What are logarithms for in real life?

Although it may seem incredible, logarithms were invented by a mathematics enthusiast named John Napier, which first appeared in his work *Mirifici logarithmorum canonis descriptio. *At that time and until the calculator was invented, logarithms were obtained through complex calculations whose results were recorded in tables.

In real life we find them present in many situations, which makes them a widely used tool in different scientific contexts, a clear example is the case of astronomers who divide stars according to their degree of luminosity (first magnitude, second magnitude, third …), to which they associate terms of an arithmetic progression: 1, 2, 3 …

For its part, the physical luminosity of these varies following a geometric progression of ratio 2.5: 2.5, 2.5 ^{2} , 2.5 ^{3} … it can be seen that the magnitude that astronomers associate with each star coincides with the logarithm of its physical luminosity in base 2.5. Which means that a fourth magnitude star is 2.5 ^{4-2} = 6.25 times more luminous than a second magnitude star. And this is just a sample of the usefulness of logarithms in real life.

## Where are logarithms used?

They can be used in various areas of knowledge, such as:

- Chemistry: to calculate the PH of substances, which is constantly measured due to the effect caused by acid rains caused by sulfur from factories and electricity plants.

- Biology: they are used to observe the nutritional effects of organisms. They are applied in the calculation of the pH which is the logarithm of the inverse of the concentrated hydrogen ions, measuring the acidity.

- Medicine: to understand certain phenomena, for example, visualize the results of the Stenberg psychological experiment based on information retrieval.

- Physics: for example, the path traveled by a ball thrown into the air, the path traveled by a river when it falls from the top of the mountain, the path of a particle with reference to the elapsed time which is thrown with an initial velocity.

- Psychology: in the case of the Weber-Fechner law (stimulus-response), which states that the response (R) is related to the stimulus (E).

- Statistics: to calculate population growth.

- In banking: they are used to measure the growth of deposits over time.

- Economy: they are applied in supply-demand, which represent indispensable relationships in any modern economic analysis.

- Music: The staff is the most common example of a logarithmic scale in which the differential in the pitch of the sound corresponds to the logarithm of the frequency.

- Civil engineering: to solve certain problems considering a support point using a second degree equation.

- Advertising: when an advertising or promotion campaign is launched, certain aspects of statistics are considered where various mathematical calculations are applied, on which the success or failure of the campaign depends to a great extent.

- Geology: they are used to calculate the intensity of an event, such as an earthquake, for example.

- Topography: to determine the height of a building when you have the base and the angle.
- Aviation: allows you to determine the distance between two planes that depart from the air base with the same speed, forming an angle and in a straight path.

## Properties of logarithms

These are the properties of logarithms:

**The logarithm of a product**is equal to the sum of the logarithms of the factors:

log _{a} (M x N) = log _{a} M + log _{b }N

It can be demonstrated by going to the definition of a logarithm:

If log _{at} M = m, and log _{at }N = n, ^{m} = M and ^{n} = N are verified

Multiplicando: M x N = a^{m }x a^{n} = a^{m+n}

Therefore, using logarithms in both expressions:

log _{a} (M x N) = m + n = log _{a} M + log _{a} N

**The logarithm of a quotient**is equal to the difference of the logarithms of dividend and divisor. The demonstration is similar to the previous one:

log _{a} (M/N) = log _{a} M _{–} log _{a} N

If log _{at} M = m, and log _{at} N = n, _{m} = M and _{n} = N are verified

Dividiendo: M/N = a^{m}/ a^{n} = a ^{m-n}

Therefore, using logarithms of both members:

log _{a} (M/N) = m –n = log _{a} M – log _{a} N

**The logarithm of a power**is equal to the exponent multiplied by the logarithm of the base:

log _{a} M ^{b} = bx log _{a} M

If log _{at} M = m, check at ^{m} = M

Raising both members of the equality to the power b: a ^{bm} = M ^{b}

Using logarithms in both members:

log _{a} M ^{b} = log _{a} (a ^{bm) =} bxm = bx log _{a} M

**The logarithm of a root**is equal to the logarithm of the radicand divided by the index of the root. The simplest proof is obtained by considering the root as a fractional power:

log _{a} M ^{1 / b} = (log _{a M} ) / b

If log _{at} M = m, it is verified at ^{m} = M

Raising both members of the equality to the power 1 / b: a ^{m / b} = M ^{1 / b}

Using logarithms in both members:

log_{a }M^{1/b }= log_{a }(a^{m/b}) = m/b = 1/b x log_{a }M

## What is a natural logarithm?

Also known as Neperian, it is the logarithm whose base is an irrational number, that is, the number *e* equivalent to approximately 2.7182818284590452353602874713527. It is denoted as In (x), log _{e} (x) or log (x) since for that number the property that the logarithm is equal to 1 applies.

Then the natural logarithm of a number x is the exponent *to* which the number *e* is raised to obtain x. The function described by natural logarithms forms an area located within a hyperbola, which is why it is called a hyperbolic function.

## What is a logarithmic function?

It is the one that is expressed as f (x) = log _{x} , which implies that the base of this function must be positive and different from 1. This function is the inverse of the exponential in the sense that:

log _{a} x = b Û a ^{b} = x

This function is characterized by the fact that its domain or initial or starting set is composed of positive real numbers, its path is R since it is a continuous function. They can be convex or concave, increasing or decreasing depending on the value of the base. In short, they are those in which the variable of the equation is the base or argument of the logarithm.

### Properties of the logarithmic function

The properties of the logarithmic function are deduced from the exponential function, being that:

- At the point x = 1, this function vanishes because in any basis log
_{a}1 = 0.

- This function only exists for positive x values (zero is not included), its domain is (0, + ∞).

- The logarithmic function of the base will always be equal to 1.

- The images obtained from the application of this function correspond to any of the elements of the set of real numbers.

- It is continuous, increasing for a> 1 and decreasing for a <1.

## Why were logarithms invented?

Initially they were called artificial numbers by their creator the Scotsman John Napier, later his name was transformed in the sense of the number that indicates a proportion. Thanks to these, complex calculations can be solved. In this sense, they are numbers that were invented or rather discovered to facilitate the resolution of geometric or arithmetic problems.

By using logarithms we can avoid all complex multiplication and division by converting it to something simpler by substituting multiplication for addition and division for subtraction. It is important to highlight the fact that the calculation of the roots is also done much more easily.

## Conclusion of logarithms

Logarithms are not simple mathematical calculations that are carried out to solve certain problems, on the contrary, they allow to simplify complex operations such as multiplication and division or raising to powers and the extraction of roots between real numbers. They are used by professionals such as astronomers, even psychologists or musicians. They are very useful in many areas where they facilitate various jobs inherent to the area.

Dr. Samantha Robson ( CRN: 0510146-5) is a nutritionist and website content reviewer related to her area of expertise. With a postgraduate degree in Nutrition from The University of Arizona, she is a specialist in Sports Nutrition from Oxford University and is also a member of the International Society of Sports Nutrition.