What is logarithm ? - Types Of Log - 10 Basic Laws Of Log

What is Logarithm? Types, Properties, Laws and Basic Formulas

Complete Guide to Logarithms, Their Types, Properties, Laws, and Applications in Mathematics

📖 Reading Time: 10–12 Minutes 🗓 Updated: July 2026 📐 Mathematics | Algebra | Logarithms

Key Takeaways

Understand the meaning of logarithms in simple language.
Learn the two major types of logarithms.
Study the fundamental laws and properties of logarithms.
Explore logarithm tables and their practical applications.
Useful for NCERT, CBSE, ICSE, JEE, NEET, Olympiads and competitive examinations.

Understanding Logarithm Types, Properties, and Fundamental Laws

Introduction

Logarithms are one of the most important concepts in mathematics and form the foundation of many advanced topics, including algebra, calculus, engineering, physics, computer science, and statistics. They simplify complicated calculations involving very large or very small numbers by converting multiplication into addition and division into subtraction.

Students often encounter logarithms for the first time in secondary school mathematics. Although the concept may appear difficult initially, it becomes straightforward once the relationship between logarithms and exponents is understood.

In this complete guide, you will learn the meaning of logarithms, their types, important properties, the ten fundamental logarithm laws, logarithm tables, practical applications, and solved examples.

What is Logarithm?

A logarithm is the inverse operation of exponentiation. It tells us the exponent to which a base must be raised to obtain a given number.

log_bx = y

if and only if

b^y = x

Here,

b = Base of the logarithm
x = Argument (the given number)
y = Logarithm or exponent

In simple words, a logarithm answers the question: "To what power must the base be raised to obtain the given number?"

Example

log₁₀100 = 2

because

10² = 100

Basic Terminology

Base

The fixed number that is raised to a power.

Argument

The number whose logarithm is being calculated.

Logarithm

The exponent required to obtain the given number.

Inverse Operation

Logarithms reverse the process of exponentiation.

Quick Facts

Logarithms are the inverse of exponential functions.
Every logarithm has a base.
The argument must be a positive number.
Logarithms simplify complex calculations.
They are widely used in mathematics, science, engineering, finance, and computer science.

Types of Logarithms

Logarithms are mainly classified according to the value of their base. Although logarithms can have many different bases, two types are most commonly used in mathematics, science, engineering, and computer science.

1. Common Logarithm

Uses 10 as its base and is widely used in school mathematics, engineering, and logarithm tables.

2. Natural Logarithm

Uses the mathematical constant e ≈ 2.71828 as its base and is widely used in calculus, higher mathematics, and scientific calculations.

1. Common Logarithm (Base 10)

A Common Logarithm is a logarithm whose base is 10. Unless the base is specifically mentioned, the notation log usually refers to the common logarithm.

log₁₀x

Since the base is always 10, mathematicians often write it simply as:

log x

Example

log₁₀100 = 2

Reason

10² = 100

Base

10

Notation

log x

2. Natural Logarithm (Base e)

A Natural Logarithm uses the mathematical constant e (Euler's Number), whose approximate value is 2.718281828...

log_ex = ln(x)

Instead of writing log_e, mathematicians usually write ln, which stands for the natural logarithm.

Example

ln(e) = 1

Reason

e¹ = e

Base

e ≈ 2.71828

Notation

ln(x)

Comparison of Common Log and Natural Log

Feature	Common Logarithm	Natural Logarithm
Base	10	e (≈ 2.71828)
Notation	log	ln
Used In	School Mathematics, Engineering	Calculus, Science, Higher Mathematics
Example	log₁₀100 = 2	ln(e) = 1

Characteristics of Logarithms

Every logarithm follows certain mathematical conditions that must always be satisfied.

Positive Argument

The number inside the logarithm must always be greater than zero.

Positive Base

The base must be positive and cannot be equal to 1.

Inverse Function

Logarithms reverse exponential operations.

Simplifies Calculations

Multiplication becomes addition, while division becomes subtraction.

Why are Logarithms Important?

Logarithms make difficult mathematical calculations much easier. Before electronic calculators became common, logarithm tables were extensively used to perform multiplication, division, powers, and roots quickly and accurately.

Major Uses

Algebra and higher mathematics.
Scientific calculations.
Engineering and physics.
Computer science and algorithms.
Financial growth and compound interest.
Earthquake magnitude (Richter Scale).
Sound intensity (Decibel Scale).

Did You Know?

The invention of logarithms by John Napier in the early 17th century revolutionized mathematics by making lengthy calculations much faster, long before modern calculators and computers were invented.

10 Basic Laws of Logarithms

The laws of logarithms are mathematical rules that simplify logarithmic expressions and make calculations easier. These identities are widely used in algebra, calculus, engineering, physics, computer science, and competitive examinations.

Important Note

The following logarithm laws are valid only when the logarithms have the same base and all arguments are positive.

1. Product Rule

The logarithm of a product is equal to the sum of the logarithms of the individual numbers.

log_a(mn)=log_am+log_an

Example:
log₂(8×4)=log₂8+log₂4=3+2=5

2. Quotient Rule

The logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.

log_a(m/n)=log_am−log_an

Example:
log₁₀(100/10)=2−1=1

3. Power Rule

The exponent of a number can be brought in front of the logarithm as a multiplier.

log_a(m^p)=p·log_am

Example:
log₂(8²)=2log₂8=2×3=6

4. Change of Base Formula

This formula converts a logarithm from one base to another.

log_ab=log_cb × log_ac

The most commonly used version is:

log_ab= log b / log a

5. Logarithm of the Base

The logarithm of the base to the same base is always equal to one.

log_aa=1

Example: log₁₀10=1

6. Logarithm of One

The logarithm of one is always zero because any positive base raised to the power zero equals one.

log_a1=0

Example: log₂1=0

7. Reciprocal Rule

The logarithm of one base with respect to another is the reciprocal of the reverse logarithm.

log_ba= 1 / log_ab

8. Exponential Identity

Logarithms and exponents are inverse operations.

a^log_am=m

9. Fractional Power Rule

When an expression contains a fractional exponent, the exponent becomes a multiplier outside the logarithm.

log_a(m^p/q)= p/q log_am

10. Equality Rule

If two logarithms with the same base are equal, then their arguments must also be equal.

log_ab=log_ac ⟹ b=c

Summary of Logarithm Laws

Law	Formula
Product Rule	logₐ(mn)=logₐm+logₐn
Quotient Rule	logₐ(m/n)=logₐm−logₐn
Power Rule	logₐ(mᵖ)=p logₐm
Change of Base	logₐb=(log b)/(log a)
Identity	logₐa=1
Zero Rule	logₐ1=0
Reciprocal Rule	log_ba=1/logₐb
Inverse Rule	a^(logₐm)=m
Fractional Power	logₐ(m^(p/q))=(p/q)logₐm
Equality Rule	logₐb=logₐc ⇒ b=c

Applications of Logarithms

Logarithms are among the most useful mathematical tools because they simplify complex calculations involving multiplication, division, powers, and roots. Today, logarithms are widely used in mathematics, engineering, physics, chemistry, computer science, finance, and many other scientific disciplines.

📐 Mathematics

Used to solve exponential equations, simplify algebraic expressions, and perform advanced calculations.

🧪 Science

Applied in chemistry, biology, and physics to analyze exponential growth and decay.

⚙ Engineering

Used in electrical engineering, signal processing, communication systems, and control engineering.

💻 Computer Science

Important in algorithm analysis, binary search, data structures, and computational complexity.

💰 Finance

Used to calculate compound interest, investment growth, and financial forecasting.

🌍 Earth Sciences

Used in earthquake magnitude (Richter Scale), sound intensity (Decibel Scale), and pH calculations.

What is a Logarithm Table?

Before electronic calculators became common, mathematicians, scientists, engineers, and students used logarithm tables to perform lengthy calculations quickly and accurately.

Definition

A logarithm table is a mathematical table that lists the logarithmic values of numbers. It helps simplify multiplication, division, powers, and roots by converting them into addition and subtraction.

Multiplication

Converted into addition using logarithms.

Division

Converted into subtraction.

Powers

Solved using multiplication of logarithms.

Roots

Calculated using division of logarithms.

Advantages of Using Logarithms

Faster Calculations

Reduces lengthy arithmetic calculations into simple addition and subtraction.

Scientific Computation

Makes handling very large and very small numbers easier.

Problem Solving

Simplifies exponential equations and complex algebraic expressions.

Higher Mathematics

Forms the foundation for calculus, differential equations, and mathematical analysis.

Common Mistakes Students Make

❌ Mistake 1

Adding logarithms without applying the Product Rule.

❌ Mistake 2

Ignoring that logarithm arguments must always be positive.

❌ Mistake 3

Confusing log with ln.

❌ Mistake 4

Using logarithm laws when the bases are different.

Remember

✔ Always check that all logarithms have the same base.
✔ The argument of a logarithm must be greater than zero.
✔ logₐ1 = 0 and logₐa = 1 are the two most frequently used identities.
✔ Practice the product, quotient, and power rules regularly.

Memory Tricks

Product

Multiplication inside → Addition outside.

Quotient

Division inside → Subtraction outside.

Power

Exponent inside → Comes in front.

Identity

Remember:
logₐa = 1
logₐ1 = 0

Exam Tips

Memorize the Product, Quotient, and Power Rules first—they are used most frequently.
Always verify that the logarithm base is the same before applying logarithm laws.
Remember that the logarithm of a negative number is not defined in real numbers.
Practice change-of-base problems regularly.
Questions on logarithm identities are common in CBSE, ICSE, JEE, NEET, Olympiads, SSC, Banking, and other competitive examinations.

Frequently Asked Questions (FAQs)

1. What is a logarithm?

A logarithm is the inverse of exponentiation. It tells us the exponent to which a base must be raised to obtain a given number.

2. What are the two main types of logarithms?

The two most common types are:

Common Logarithm (Base 10)
Natural Logarithm (Base e)

3. What is the difference between log and ln?

log generally represents the logarithm with base 10, whereas ln represents the natural logarithm whose base is e ≈ 2.71828.

4. Can a logarithm have a negative argument?

No. In the real number system, the argument of a logarithm must always be greater than zero.

5. Why is log_a1 equal to 0?

Because every positive number raised to the power of zero equals one.

a⁰=1

Therefore, log_a1=0

6. Who invented logarithms?

Logarithms were introduced by the Scottish mathematician John Napier in the early seventeenth century to simplify lengthy mathematical calculations.

7. Where are logarithms used?

Logarithms are used in mathematics, engineering, physics, chemistry, computer science, finance, statistics, earthquake measurement (Richter Scale), sound intensity (Decibel Scale), and pH calculations.

8. Why are logarithm tables important?

Before electronic calculators became common, logarithm tables enabled fast multiplication, division, powers, and roots by transforming complicated calculations into simple addition and subtraction.

Quick Revision

✔ A logarithm is the inverse operation of exponentiation.
✔ log_bx = y means b^y = x.
✔ The two main types are Common Logarithm and Natural Logarithm.
✔ Product Rule converts multiplication into addition.
✔ Quotient Rule converts division into subtraction.
✔ Power Rule brings the exponent in front of the logarithm.
✔ The logarithm argument must always be positive.
✔ log_aa = 1 and log_a1 = 0 are fundamental identities.

Summary Table

Concept	Description
Definition	Inverse of exponentiation.
General Formula	log_bx = y ⇔ b^y = x
Common Logarithm	Base 10
Natural Logarithm	Base e
Most Important Laws	Product, Quotient, Power, Change of Base
Conditions	Base > 0, Base ≠ 1, Argument > 0
Main Applications	Mathematics, Engineering, Physics, Finance, Computer Science

Conclusion

Logarithms are one of the most powerful mathematical concepts because they transform complex exponential calculations into simple arithmetic operations. By understanding the relationship between logarithms and exponents, students can solve a wide variety of mathematical problems more efficiently.

The fundamental laws of logarithms—including the Product Rule, Quotient Rule, Power Rule, and Change of Base Formula—form the foundation of algebra, calculus, engineering, physics, statistics, computer science, and many scientific applications. Mastering these identities not only improves problem-solving skills but also prepares students for higher-level mathematics and competitive examinations.

Key Formula

log_bx = y

if and only if

b^y = x

References

NCERT Mathematics Textbook (Class XI)
NCERT Exemplar Problems – Mathematics
CBSE Mathematics Curriculum
John Napier – Mirifici Logarithmorum Canonis Descriptio
Hall & Knight – Higher Algebra
Standard Algebra and Calculus Textbooks

Tags: Mathematics