# Applications of Logarithms

Logarithms have a large number of applications in real life. Logarithms are especially used to create measurement scales that are more manageable. Some examples of the applications of logarithms include the Richter scale for measuring earthquakes, the decibel scale for measuring sound, orders of magnitude, and applications in data analysis.

Here, we will look at these applications in more detail.

##### ALGEBRA

Relevant for

Learning about various applications of logarithms.

See applications

##### ALGEBRA

Relevant for

Learning about various applications of logarithms.

See applications

## Orders of magnitude

When we express something like “a 6 digit salary”, we are describing the numbers depending on how many powers of 10 they have (they are in the tens, hundreds, thousands, etc). Adding a digit means multiplying by 10. For example, 1 has one digit and 100,000 has six digits.

Logarithms count the number of multiplications that are added to obtain a number, Therefore, starting with 1 (a single digit), we add 5 more digits ($latex {{10}^5}$) and we obtain 100,000, a six-digit number. Logarithms help us represent numbers using more manageable scales. It is easier to talk about something that has 6 digits than to mention that we have a hundred thousand.

In computers, where everything is counted with bits (1 or 0), each bit has a doubling effect (not × 10). Thus, if we go from 8 bits to 16 bits, this is 8 orders of magnitude or $latex {{2}^8}=256$ times larger. Changing from 16 bits to 32 bits represents a change of 16 orders of magnitude or $latex {{2}^{16}}=65536$ times larger.

## Richter scale

The Richter scale is a base-10 logarithmic scale. This scale defines the magnitude of an earthquake as the logarithm of the ratio of the amplitude of seismic waves to an arbitrary standard amplitude:

$latex M = \log (\frac{A}{S})$

where, A is the amplitude of the earthquake measured with a seismometer from approximately 100 km from the epicenter of the earthquake and S is the standard amplitude of an earthquake, which is defined as approximately 1 micrometer.

Since the Richter scale is a base 10 logarithmic scale, each increment of one on the Richter scale indicates an intensity ten times stronger than the previous number on the scale.

## Decibel scale

Sound carries energy and its intensity is defined as:

$latex I =\frac{P}{A}$

where, P is the power, which indicates the energy that flows per unit area, A, which is perpendicular to the direction in which the sound wave travels.

Sound intensity is measured in terms of volume, which is measured in terms of a logarithm. Therefore, the intensity of the sound is defined as:

$latex \beta = (10dB) \log (\frac{I}{I_{0}})$

In this definition, dB represents decibels which are equal to one-tenth of a bel (B). I is the intensity of the sound and $latex I_{0}$ is the standard intensity.

With decibels, we can represent sound intensities that vary greatly in magnitude on the same scale.

## Applications in data analysis

Logarithms are widely used in data analysis, which in turn is used in data science and computational machine learning.

The logit plays a very important role in logistic regression. All probabilities can be easily converted to logit.

Logarithmic transformations are also important to make it easier to see patterns in your data. Using logarithmic transformations it is possible to obtain exponential functions that are easier to read and are more understandable.

Since logarithms can model a wide variety of phenomena, they are extremely useful in data science. Much of data science is modeling real-life situations, so logarithmic scales are vital.

Google gives every page on the web a score (PageRank), which is roughly a measure of the website’s authority and importance of the page. This is a logarithmic scale, which means that PageRank counts the number of digits in the score.

For example, a site with a PageRank of 2 (2 digits) is 10 times more popular than a site with a PageRank of 1. CNN’s PageRank is 9, so there is a difference of 4 orders of magnitude ($latex {{ 10}^4}=10000$) compared to a page with a PageRank of 5.