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Step-by-Step Guide to Calculate Antilog

In mathematics, antilogarithms (or antilogs) are a fundamental concept that often comes across in scientific calculations, engineering, and various applied fields. Understanding how to calculate antilogarithms is important for anyone working with logarithms or exponential functions. 

This article aims to guide you through the process of calculating antilogarithms step-by-step, ensuring a clear and comprehensive understanding of the concept.

What is Antilog?

The antilogarithm commonly called antilog, is the mathematical operation that undoes the logarithm. It's a fine function that helps us find the original value from a given logarithmic value. In simpler terms, the antilog allows us to determine the actual number when we only have the logarithm of that number.

The formula:

Antilog(x) = b^x.

Here, 'x' represents the logarithmic value, and ‘b’ represents the base.

How do we calculate the Antilog?

Understanding the parts of the logarithm mantissa and its characteristics is important to find the antilog of a number. Let's explore each part:

1. Characteristic: The characteristic is an integer part of the logarithm. In a logarithm expression like log(x), the characteristic represents the whole number part of the result.
2. Mantissa: The decimal part of the logarithm is the mantissa. It is the fractional portion following the characteristic. In a logarithm expression like log(x), the mantissa represents the decimal value that gives the complete logarithm when combined with the characteristic.

When finding the antilogarithm, separate the characteristic and mantissa parts.

• The characteristic helps identify the number of digits before the decimal point in the antilogarithm result.
• The mantissa helps determine the decimal value.

Steps:

To calculate the antilogarithm, follow these steps:

1. Identify the characteristic and mantissa from the logarithm value.

2. Compute the antilogarithm of the mantissa

3. Adjust for the characteristic: Move the decimal point in the antilogarithm result depending on the characteristic value. 

1. If the characteristic is positive:

1. Count the number of digits in the characteristic.

2. Move the decimal point in the antilogarithm result to the right by the same number of digits.

3. If the antilogarithm result is a whole number, add zeros as necessary after moving the decimal point.

2. If the characteristic is negative:

1. Take the absolute value (positive value) of the characteristic.

2. Count the number of zeros before the first non-zero digit in the antilogarithm result.

3. Move the decimal point in the antilogarithm result to the left by the same number of zeros.

4. If the antilogarithm result is less than one, add zeros as necessary before moving the decimal point.

Manual calculation of antilogarithms can be time-consuming and prone to errors. To simplify the process, consider using Antilog Calculator by Allmath . This tool is useful for anyone dealing with complex logarithmic data. Enter your logarithm value and base in the provided box and click the calculate button.

How do we use an Antilog Table?

An antilog table provides calculated values of antilogarithms for a range of logarithmic values, it's also known as a logarithm table. To avoid the complex calculations this table helps us. 

To use an antilog table:

1. Locate the characteristic and Mantissa value in the table.
2. Focus on the mantissa. Use the first two digits after the decimal point as the row number and the third digit as the column number in the antilog table.
3. Look for the mean difference in the same row corresponding to the fourth digit of the mantissa. Add this mean difference to the value obtained in Step 2.
4. Place a decimal point right after the first digit obtained in Step 3.
5. Multiply the number obtained in Step 4 by 10 raised to the power of the characteristic. The result is the antilog of the given number.

Properties of Antilog

Antilogarithms possess several interesting properties that are useful in various mathematical calculations. Some of the key properties of anti-logarithms include:

1. Identity property: The algorithm of 0 is always 1.

2. Inverse property: antilog(log(x)) = x log(antilog(x)) = x.

3. Product property: antilog (x + y) = x log (antilog (x)) = x.

4. Quotient Property: (x - y) = antilog (x) / antilog (y).

These properties enable us to perform various calculations and simplify complex mathematical expressions involving antilogarithms.

Applications of Antilog

Antilogarithms have numerous applications in various fields, including mathematics, physics, engineering, finance, and scientific research. Some noteworthy uses of anti-logarithms are:

1. Exponential Growth and Decay: Antilogarithms aid in modeling exponential growth and decay phenomena. They are used to calculate the values at different time points based on growth rates or decay rates.

2. Signal Processing: Antilogarithms are utilized in signal processing algorithms to convert logarithmic measurements back into linear scale values. This conversion is crucial in applications such as audio and image processing.

The versatility of antilogarithms allows for their extensive application across diverse scientific and technical domains.

Example of Antilog

This section will demonstrate how to calculate antilog with the help of examples.

Example 1: (with table)

Calculate the antilog of 2.4856 

Solution:

Using an antilog table.

Step 1:

Locate the characteristic and Mantissa value in the table.

Characteristic: 2

Mantissa: 0.4856.

Step 2:

Focus on the mantissa. Use the first two digits after the decimal point 48 as the row number and the third digit 5 as the column number in the antilog table.

Row number = 48

Column number = 5

Corresponding value = 3055 + 4

Step 3:

Look for the mean difference in the same row corresponding to the fourth digit of the mantissa. Add this main difference of 4 to the value obtained in Step 2.

 = 3055 + 4 = 3059.

Step 4:

Place a decimal point right after the first digit obtained in Step 3:

 = 3.059.

Step 5:

Multiply the number obtained in Step 4 by 10 raised to the power of the characteristic. The result is the antilog of the given number.

= 3.059 × 103 = 305.9

Hence, the antilog of 2.4856 is 305.9.

Example 2: (without table)

Find antilog of 2.4856 without table.

Solution:

Step 1: 

Identify the characteristic and mantissa from the logarithm value.

Characteristic: 2

Mantissa: 0.4856

Step 2:

Compute the antilogarithm of the mantissa.

Using the exponential function, calculate the antilogarithm of the mantissa value:

Antilogarithm of 0.4856 = 10(0.4856)

Step 3: 

Adjust for the characteristic.

Since the characteristic is 2, move the decimal point two places to the right.

Multiply the antilogarithm result by 100:

Adjusted antilogarithm = Antilogarithm of 0.4856 × 100

Now, we need to evaluate the exponential function using a calculator or mathematical software:

Antilogarithm of 0.4856 ≈ 3.0445

Step 4: 

Adjusted antilogarithm = 3.0445 × 100

Multiply 3.0445 by 100:

Adjusted antilogarithm ≈ 304.45.

Conclusion

In this article, we explored the concept of antilogarithm, its calculation method, properties, and applications, and provided examples. Antilog is essential in undoing logarithms and finding the original values. Whether using an antilog table or mathematical calculations, understanding the characteristics and mantissa helps in accurate results. Antilog finds applications in various fields like exponential growth, signal processing, and more.

What is Mathematics? Definition, Branches, Books and Mathematicians

Hello Friends! I hope you're having a great time reading my articles. Today, we are going to learn about a very comprehensive topic What is Mathematics? Do you have an interest in math? Great! This tutorial will surely help you to know about Mathematics definition, branches of Mathematics, the importance of mathematics in our life, famous books of mathematics and popular mathematicians & their discoveries. The role of math's in our daily life is like a building block (essential), involves in our every regular activities. i.e. engineering, mobile devices, businesses etc. Even math is there during our sports time, if you ever noticed. Let's take the example of our body where protein (amino-acid) is the building block. Without protein, our body can't exist. Similarly, math plays the same role in every field of science. For instance, without math, engineers can't build great buildings and scientists can't develop advance machines. So, think about it for a second! Let's move on & further discuss what is mathematics?

What is Mathematics?

Do you want to know why the need for mathematics arose? Math actually came because of the need of society. In ancient times, math was not that complex. The tribes use maths only for counting. With time they also start relying on math to calculate sun position. As you know, our needs have become more complex, thus math also becoming complex day by day.

Mathematics Definition

• Mathematics is the branch of science, which deals with numbers, involves calculations and mainly focuses on the study of quantity, shapes, measurements etc.
• The greatest mathematician Benjamin Peirce defined math as “the science that draws the necessary conclusion”.
• In more simple words, math is the science, deals with structures, numbers, geometry etc.

Why study mathematics?

In past, the study of math was very limited. And only 3 fundamental branches of mathematics were discovered. These were geometry, algebra and number theory. But in the modern age, several other branches have been discovered by mathematicians based on the main branches. We will discuss these branches later in this article. Here the point is that now mathematics has become a very vast and most discuss field of science. Now, mathematics got the title of “queen of science”. Math became now an inseparable part in our everyday life tasks. Finally, let me conclude that mathematics is a vital tool in every field of science throughout the world. Such as:

• Natural science.
• Engineering.
• Medicine.
• Social sciences.

Importance of Mathematics

Let's see why mathematics is important for us. I will discuss some common examples of everyday routine. I must say, our day starts with math.

• During cooking food in the kitchen, we are able to measure every ingredient, just because of math.
• Going on shopping & paying cash involves mathematics.
• The hobbies like arts, gardening & playing, all need mathematics.
• Texting on phones & faxing also require math

Branches of mathematics

Just like other fields of science, mathematics is also divided into the following branches. So, let us discuss them briefly.

1. Arithmetic

This is most oldest and fundamental branch of math. This branch deals with the basic operation & number. The basic operations are subtraction, addition, multiplication and division.

2. Algebra

Algebra is the type of arithmetic. Here, we find the value of unknown quantities, such as X,Y and Z. we usually used English alphabets (variables) as unknown numbers. Several formulas uses to solve the equations to calculate the unknown value. You must have solved algebra questions in high school education.

3. Geometry

The next branch we will discuss is geometry, the most practical branch considered. It deals with the construction of figures, shapes and their properties. The basics of geometry are points, lines, surfaces, angles and solids.  In geometry, we used many mathematics tools such as scale & protector.

4. Trigonometry

Trigonometry is derived from Greek words trigon means triangle and metron refer to measurement. So, it is clear from the name that is the study of triangle sides and angles. You will get to learn this branch in higher education.

5. Analysis

This is a little complex branch of mathematics. It deals with the study of rate of change with respect to specific quantities. The base of analysis is Calculus.

6. Statistics

The statistic is the branch, deals with the collection of huge data, organize it and further analyze to get final results. Such as the population of any country estimated through this branch.

Applications of Mathematics

• Forecasting the Weather. Mathematics helps in predicting one of the most difficult task on earth.
• Reading of CDs and DVDs.
• MRI & Tomography.
• Use in Internet and Phones.
• Analysis of Epidemics.
• The estimation of glacier melting.
• Maps of the Earth.
• Cryptography.

Popular Mathematicians

The role of greatest mathematicians in our life is remarkable. These are the mathematicians who made it possible to transfer electricity to the distance of thousands of kilometer enhance the computer technology, helps to reveal the DNA structure and still helping the scientist to explore the universe deeply. Let's get to know the contributions of these great mathematicians.

The Greatest Mathematicians of all Times
No.	Mathematicians	Contributions
1	Srinivasa Ramanujan	He was one of the greatest mathematicians died at early age of 32. He calculated Euler–Mascheroni constant at the age of 16. He identified more than 4000 math identities.
2	Joseph-Louis Lagrange	He was known for his discoveries i.e. the number theory, Lagrangian mechanics and Celestial Mechanics. He also had remarkable contribution in the formation of Euler–Lagrange equation. Moreover, he is famous for his invention in analytical mechanics, helped the researchers to develop mathematical physics branch.
3	Andrew Wiles	He is a famous British mathematician, won Wolf Prize in 1995 and Abel Prize recently in 2016.He is popular for the formulating the Fermat’s Last Theorem. That was one of the complex problems in mathematics.
4	Alan Turing	He was known for his contribution such as Turing’s proof and Cryptanalysis of the Enigma. He won the Smith’s Prize in 1936.In second world war, his contribution was outstanding. It was him, whose cryptanalytic abilities improved the bombe and developed a faster decoding Enigma machine.
5	G.F. Bernhard Riemann	He is famous for his Fourier series and Riemann integral series. He worked on  differential geometry, invent his own theory on higher dimensions. His contributions, known for Riemannian geometry. He was the founder of the Riemann mapping theorem.
6	David Hilbert	He was famous for his Hilbert’s problems and proof theory. He was the greatest mathematician, developed instruments in the field on commutative algebra. He had also contributed in the field of calculus (variations) & mathematical physics. He also worked in the field of physics.
7	Bernoulli Family	Bernoulli family had a special respect in mathematics. Johann and Jacob from Bernoulli family, were the first mathematician worked for calculus and invented Bernoulli numbers & Brachistochrone curve.
8	Issac Newton	He was known for his Newton’s laws of motion as well as Newtonian mechanics & Calculus. He was considered as the father of mechanics. His thought and discoveries about gravitation force are accepted universally.  He was the founder of Einstein’s theory of relativity.
9	Pythagoras	He invented Pythagorean theorem as well the founder of Theory of Proportions. The Pythagoras theorem was named after him.
10	Carl Friedrich Gauss	He  won the Lalande Prize in 1809 and Copley Medal 1838.He was one of the most influential mathematicians in the ancient Greek. He worked for many mathematicians branches as well in physics. He has ability to solve arithmetic problems faster than anyone. Some of his inventions are Gauss’ Law & Theorema Egregium. He also estimated the non-Euclidean geometry.

List of popular Mathematicians

There are names of many other great mathematicians, I am goin to enlist here.

• Plato.
• Euclid.
• Eratosthenes.
• Hipparchus.
• Hypatia.
• Girolamo Cardano.
• Leonhard Euler.
• Carl Friedrich Gauss.
• Georg Cantor.
• Paul Erdös.
• John Horton Conway.
• Grigori Perelman.
• David Harold Blackwell.
• Jesse Ernest Wilkins.
• M. Euphemia Lofton Haynes.
• Joseph James Dennis.
• Wade Ellis.
• Clarence F. Stephens.
• Evelyn Boyd Granville.
• Marjorie Lee Browne.
• Georgia Caldwell Smith.
• Gloria Conyers Hewitt.
• Mary Rodriguez.
• Thyrsa Frazier Svager.
• Vivienne Malone-Mayes.
• Shirley Mathis McBay.
• Eleanor Green Dawley Jones.
• Geraldine Claudette Darden.
• Annie Marie Watkins Garraway.

Popular Mathematics Books

Reading good books can open your mind to accept new light. There are so many math's good books available out there. Let's discuss must read mathematics books.

1. The Art of Statistics

This book is a comprehensive study about the stat tools. This book guide you how to use statistics formulas and tools properly. It also reveled the areas of study where stat can be used. I have read this book, and found it very useful to understand statists problems.

2. Do Dice Play God?

I really found this book rewarding, tells us about the basics of quantum mechanics. The author explains the complex problems of quantum theory deeply, even I have never read so deep guidance in other popular science books. Ian Stewart has already written so many math books. But this one is the must read math book.

3. Humble Pi

This is the most interesting book i ever read. Actually the author is the comedian, and he wrote math problems in a very funny way to attract the readers. But on a serious note, this book also solve many issues you will commonly found in math. So this is my third favorite math book.

The list is going on. As there are countless valuable mathematics books published. I couldn't stop

myself from only suggesting three books. So, let's get the name of few more popular books of mathematics.

List of top mathematics books highly recommended

• Encyclopedia of Mathematics. By James Stuart Tanton.
• The four pillar of Geometry. By John Stillwell.
• The Maths of Life and Death.
• A Mathematical Introduction to Logic, Second Edition by Herbert Enderton.
• Calculus made easy by Silvanus P. Thompson.
• Introductory Statistics by Neil A. Weiss.
• Introduction to Algorithms, Third Edition by Thomas H. Cormen, Charles E. Leiserson and Ronald L. Rivest
• Categories for the Working Mathematician by Saunders Mac Lane.
• Principles of Mathematical Analysis, Third Edition by Walter Rudin.
• The Calculus Lifesaver: All the Tools You Need to Excel at Calculus by Adrian Banner
• Linear algebra done right by Sheldon Axler.
• Elementary number theory by Gareth A. Jones and Josephine M. Jones.
• Introduction to Topology and Modern Analysis by George F. Simmons.
• Abstract Algebra by David S. Dummit and Richard M. Foote.
• Basic Mathematics by Serge Lang.

I hope this article will help to get better understating of mathematics, its branches, popular books and mathematicians.

Introduction to Quadratic Equations with it's Graphical Representation

Welcome everyone! In this article, we will have a detailed Introduction to Quadratic Equations. Quadratic equations are simple second degree polynomials, but because of their extensive application, they have been assigned a special name Quadratic and scientists (over a period of time) have designed numerous methods to solve Quadratic Equations. So, today we not only cover the Quadratic Equations but will also have a look these methods & their implication. Here's a summary, which we will be covering today:

• Introduction to Quadratic Equations.
• Solutions of a Quadratic Equation.
• Graphical Representation of Quadratic Equations.
• Methods to solve Quadratic Equation.
• Multiple Graphical Solutions of Quadratic Equations.
• Forms of  Quadratic Equations.
• Comparison between quadratic equations.

So, let's get started with detailed Introduction to Quadratic Equations:

Introduction to Quadratic Equations

When we talk about numbers , it is quite obvious to think about their combination, which are actually polynomials having different degrees. You may think why we are discussing polynomials here while we are interested in quadratic equations, you’ll  get this quadratic equation from polynomials actually.  Since to make some combination we need some constants and variables, so for some constant a  and some variable ‘x’, we can write ‘ax’ which is a product of constant a and variable ‘x’.  Now if we add one more constant   by writing in a way such that ‘ax+b’, so this is a polynomial of  degree 1  , because here power of ‘x’ is 1 and we can call it a linear equation.  Similarly a second degree polynomial will be ‘ax2+bx+c’ with a,b,c constants and you may consider these constants are real numbers. Generally a polynomial of degree n can be written as below:

a0+a1x+a2x2+ . . . +anxn

where all a0, a1, a2, . . . ,an are constants which belong to the set of real numbers. Here we will just talk about 2nd degree polynomial. A 2nd degree polynomial of the form      ‘ax2+bx+c’  is called a quadratic polynomial and by equating equal to 0 we get a quadratic equation, which is:

ax2+bx+c=0 where a,b & c are constants and real numbers.

Here a is not equal to 0, otherwise it will be a linear equation. As a quadratic equation has the form  ax2+bx+c=0  , where a is necessarily not equal to 0, b and c may be 0. So an equations of the form   ax2+c=0  and ax2+bx=0  are also quadratic equations having different graphical representations.

Hitory of Quadratic Equations

After studying simple linear equations, mathematicians put their minds towards 2nd degree equations. The Egyptian Mathematician, Berlin Papyrus, gave the idea of a two-term quadratic equation. After that, Chinese mathematicians used geometric methods to solve quadratic equations with positive roots, by defining on the real line.

Possible Solutions of a Quadratic Equation

As in the quadratic equation, we have highest degree 2 of a term , known as quadratic term and shows that this equation may have at most two solutions. These two solutions of a quadratic polynomial are called zeros or positive roots of the equation. Some times we can find both solutions easily but some times it is hard to find exactly 2 solutions and in that case root does not lie on the real line. We may get 0, 1 or 2 solutions of a quadratic equation.

Solutions of a quadratic equation

In general, there are only  2 solutions exist for a quadratic equation because it is a 2nd degree polynomial and are named as positive roots.

Graphical Representation of Quadratic Equation

Yes! You can analyze Quadratic Equations graphically. Quadratic equations represent a parabola, if it meets at some points on the real line then those points are roots of the equation, otherwise it has no solution. Here the following figure is showing a graph of quadratic equation. As in the above figure we can see that a parabola on the right side with yellow colour meets at point 1 and 6 on the real line so these two points are the roots and solution of this quadratic equation. Well , it’s not always possible to draw all the solutions graphically. You can see from the above figure of parabola having pink color does not meet at any point on the real line so it means that parabola has no solution.

How to Solve Quadratic Equations ?

Now, let's have a look at How to solve quadratic equations and get its roots (if exist). There are different methods to solve these quadratic equations and here I am going to discus three of them, which are most commonly used.

1. Quadratic Formula

• We can find solutions/roots of a quadratic equation by using simple a well known formula which is known as quadratic formula and is given below:

2. Method of Factorization

• Secondly, we can solve quadratic equation by another method which is known as method of Factorization. This method is more clear from the following figure which shows step by step procedure to apply on some quadratic equation.

3. Method of Completing Square

• We can find roots of a quadratic equation by using method of Completing Suqare.

Comparison between Quadratic Equations

Suppose you have 2 different quadratic equations x2+x+-12=0 and the other equation  3x2+3x+-36=0 and you want to compare these equations. To check we must have to focus on their roots and graphical representation , let’s solve this equation by using method of factorization.

3x2+3x+-36=0              ,              x2+x+-12=0           

3(x2+x+-12)=0               ,             x2+4x-3x+-12=0      

 x2+x+-12=0                  ,              x (x+4) – 3 (x+4) =0

                                                      (x-3) (x+4)=0   and this shows x=3 and x=-4.

From both above equations, we see that both equations have same roots but we cannot say these both equations are exactly equal, because we are not sure about all those points of the parabola other than roots. But by drawing parabolas of these equations we can judge easily and the following figure is showing these parabolas have only those common points  which are their roots. So we cannot say these are equal quadratic equations because their behaviour is not same graphically. From here, we can also say that if roots are same for some quadratic equations then it doesn’t mean all those equations will be the same.    So, that was all for today. I hope you have enjoyed today's lecture. If there's some issues, let me know in comments and I will try to resolve them. Thanks for reading. Have a good day !!! :)