スキップしてメイン コンテンツに移動

Understanding Logarithmic Functions from Scratch

Logarithmic functions are a powerful mathematical tool, acting as the inverse of exponential functions, allowing us to replace multiplicative relationships with additive ones and understand the fluctuations in the magnitude of numbers.

When dealing with the rapid growth or decay of exponential functions, logarithmic functions play a role in compressing the scale of data and making complex phenomena easier to understand.

This article explains what logarithmic functions are, where they are used, and the benefits of learning them.

1. What are Logarithmic Functions?


A logarithmic function indicates the inverse operation of a positive real number raised to a power.

- Inverse of Exponential Functions

Logarithmic functions perform the inverse operation of exponential functions to derive values.

- Scale Transformation

Taking the logarithm transforms multiplicative relationships into additive ones. This makes it easier to handle very large values or rapid changes.

2. Where are they Used?


Logarithmic functions are practically applied in a wide range of fields, leveraging their properties. Here are some main examples:

- Sound Decibels (dB)

The loudness of sound takes a very wide range of values, but logarithmic transformation converts it into a more manageable scale (decibels).

- pH Scale

The acidity or alkalinity of a solution is expressed using the logarithm of the hydrogen ion concentration, allowing for intuitive evaluation.

- Earthquake Magnitude

The Richter scale evaluates earthquake energy logarithmically, making even small differences significant.

- Compound Interest

Asset growth increases exponentially due to the compounding effect. Logarithms allow you to represent growth on a linear graph, making future predictions easier.

- Economic Growth Analysis

By logarithmically transforming large numerical fluctuations, it becomes easier to analyse trends, periodicity and outliers in economic data.

- Information Theory

The evaluation of information quantity, such as entropy, is often calculated using logarithmic functions. This is very useful for quantifying data uncertainty and the spread of distribution.

- Data Normalisation

By logarithmically transforming very large values or exponential data, you can shape the data into a form suitable for linear models, leading to improved accuracy in machine learning models.

3. What are the Benefits of Learning?


There are many benefits to learning logarithmic functions, not only for academic foundations but also for practical applications in society.

- Deepening Mathematical Understanding

Exponential and logarithmic functions are fundamental concepts in a wide range of mathematical fields, including calculus, linear algebra, and probability. Understanding these opens the door to more advanced mathematical theories.

- Improving Data Analysis Skills

Logarithmic transformation is a powerful tool for adjusting data scale and simplifying complex phenomena. This makes data pre-processing and modelling easier, and improves analysis accuracy.

- Practical Application Skills

In fields such as finance, engineering, and information science, knowledge of logarithmic functions is immediately useful. The ability to perceive real-world problems in terms of exponential changes and multiplicative relationships is a significant advantage in business and research.

- Improving Problem-Solving Skills

By using logarithmic functions, complex multiplication or exponential problems can be converted into addition or linear problems, broadening the scope of thought when finding solutions.

In Summary


Logarithmic functions are a mathematical tool that, as the inverse of exponential functions, effectively expresses rapid growth or decay. 

Learning logarithmic functions will allow you to more simply understand phenomena with rapid fluctuations and big data, and your mathematical and analytical skills will be dramatically improved. 

If you want to learn logarithmic functions, we recommend this book (click here).




コメント

コメントを投稿

このブログの人気の投稿

Understanding Probability and Probability Distributions from Scratch

 In modern society, we are surrounded by various uncertainties and random phenomena. From the weather and stock prices to the outcomes of sports and even small daily choices, the concepts of probability and probability distributions are powerful tools for understanding these uncertainties quantitatively. This article explains what probability and probability distributions are, where they are used, and the benefits of learning these concepts. 1. What are Probability and Probability Distributions? Probability is a way of expressing the likelihood of an event occurring as a number between 0 and 1. 0 means the event will not occur, and 1 means the event will definitely occur. The mathematical thinking behind probability is often subtly present when we talk about the “likelihood” of something happening in everyday life. A probability distribution systematically represents all possible outcomes and the probability of each outcome. - Discrete Probability Distribution This applies to distr...
コメントを投稿
続きを読む

Entendiendo la Regresión de Bosques Aleatorios desde Cero

En el panorama actual de la ciencia de datos, los algoritmos capaces de manejar eficazmente relaciones no lineales e interacciones complejas están muy demandados. Entre estos, la Regresión de Bosques Aleatorios destaca como una técnica flexible y potente, logrando una alta precisión predictiva al combinar numerosos modelos de regresión de árboles de decisión. Este artículo explica los conceptos básicos de la Regresión de Bosques Aleatorios, los escenarios donde sus fortalezas se utilizan mejor y los beneficios de aprender esta técnica. 1. ¿Qué es la Regresión de Bosques Aleatorios? La Regresión de Bosques Aleatorios es una técnica de regresión que integra múltiples modelos de regresión de árboles de decisión en forma de “aprendizaje conjunto” (ensemble learning). – Principios Básicos Cada árbol de decisión se construye utilizando muestras bootstrap (remuestreo de los datos) del conjunto de entrenamiento. Además, las características utilizadas para la división en cada nodo se selecciona...
コメントを投稿
続きを読む

Understanding Differential Equations Solved with Variation of Parameters

1. What are Differential Equations Solved with Variation of Parameters? Differential equations are a powerful tool for mathematically capturing changing phenomena. Among these, the “method of variation of parameters” is a particularly useful technique for solving non-homogeneous linear differential equations. The general solution to a homogeneous differential equation is known, expressed by a combination of constants (constant coefficients).  However, this cannot be directly solved when a non-homogeneous term (corresponding to an external influence or input) is added. Therefore, the method of variation of parameters takes an approach of replacing the original constant parts with (unknown) functions and determining the shape of those functions through differentiation. This method allows the construction of a complete solution including the non-homogeneous term.  Due to its flexibility in handling various systems – such as when the non-homogeneous term is an exponential function...
コメントを投稿
続きを読む