Common Logarithm – Definition, Formula, Examples and Applications

The common logarithm is one of the most important concepts in mathematics. It is widely used in algebra, science, engineering, statistics, finance, and computer science. Understanding the common logarithm makes it easier to solve exponential equations and perform calculations involving very large or very small numbers.

In these notes, you will learn the definition of the common logarithm, its formula, properties, examples, and practical applications in an easy-to-understand student-friendly style.

What is a Common Logarithm?

A common logarithm is a logarithm whose base is 10.

If

Since base 10 is used so frequently, mathematicians usually write

log (N)

instead of

Definition

The common logarithm of a positive number is the exponent to which 10 must be raised to obtain that number.

Base of Common Logarithm

The base of common logarithm is always 10

Examples:

log 10=1

because

log 100 = 2

because

log 1000 = 3

because

Common Logarithm Formula

The basic common logarithm formula is

log N = x

if

where

• Base = 10
• N = Positive number
• x = Logarithm of N

Common Logarithm Properties

Product Rule

log(AB) = log A + log B

Example

log(100 * 1000) = log100 + log1000

i.e. 2 + 3 = 5

Quotient Rule

log(A/B) = log A – log B

Example

log(1000/100) = 3 – 2 = 1

Power Rule

Example

Logarithm of One

log1 = 0

because

Logarithm of Base

log10 = 1

Examples

Example 1

Find

log100

Solution

Since

Therefore,

log100 = 2

Example 2

Find

log10000

Solution

Since

Therefore,

log10000 = 4

Example 3

Find

log0.01

Solution

Since

Therefore,

log0.01 = -2

Example 4

Find

log100000

Solution

Since

Therefore,

log100000 = 5

Common Logarithm Calculator

A common logarithm calculator is an online tool that calculates the logarithm of a positive number with base 10 instantly.

Students can use a calculator to:

• Verify manual calculations.
• Solve lengthy numerical problems.
• Check homework answers.
• Learn logarithmic concepts more efficiently.

Although calculators are useful, students should first understand the definition and formula before relying on them.

Applications of Common Logarithm

The common logarithm has many practical uses.

Science

Scientists use logarithms to represent very large and very small quantities.

Engineering

Engineers use logarithms in electrical circuits and signal processing.

Earthquake Measurement

The Richter Scale uses logarithmic calculations to measure earthquake intensity.

Chemistry

The pH scale is based on logarithms.

Finance

Logarithms help calculate compound growth and investment returns.

Computer Science

Algorithms and computational complexity often involve logarithms.

Important Points to Remember

• The common logarithm always has base 10.
• Only positive numbers have real logarithms.
• (log1 = 0)
• (log10 = 1)
• Logarithms convert multiplication into addition.
• They also convert division into subtraction.

Conclusion

The common logarithm is one of the most useful mathematical concepts for solving exponential problems and simplifying calculations. By learning the definition, formula, properties, and applications of common logarithms, students build a strong foundation for higher mathematics, science, and engineering.

Short Notes

Common Logarithm (Short Notes)

• A common logarithm is a logarithm with base 10.
• It tells the exponent to which 10 must be raised to obtain a given number.
• Formula:

log N = x

iff

• Base of common logarithm = 10
• (log1 = 0)
• (log10 = 1)
• Product Rule:

• Quotient Rule:

log(A/B) = log A – log B

• Power Rule:

• Used in science, engineering, chemistry, finance, and computer science.

MCQs

1. The base of the common logarithm is:

A) 2

B) e

C) 10

D) 5

Answer: C

2. What is (log100)?

A) 1

B) 2

C) 3

D) 10

Answer: B

3. What is (\log1)?

A) 0

B) 1

C) −1

D) 10

Answer: A

4. Which formula is correct?

A) log(AB) = log A – log B

B) log(A+B) = log A + log B

C) log(AB) = log A + log B

D) None of these

Answer: C

5. What is (log1000)?

A) 2

B) 3

C) 4

D) 5

Answer: B

6. Which number has a common logarithm of 4?

A) 100

B) 1000

C) 10000

D) 100000

Answer: C

7. Which of the following numbers has a negative common logarithm?

A) 100

B) 10

C) 0.1

D) 1000

Answer: C

8. The common logarithm is mainly used with which base?

A) 5

B) 8

C) 10

D) 12

Answer: C

9. Which field commonly uses logarithms?

A) Chemistry

B) Engineering

C) Finance

D) All of these

Answer: D

10. Logarithms help convert multiplication into:

A) Division

B) Addition

C) Subtraction

D) Powers

Answer: B

Worksheet / Assignment

Part A: Fill in the Blanks

1. The base of the common logarithm is ________.
2. (log1 = ) ________.
3. (log100 = ) ________.
4. (log1000 = ) ________.
5. Common logarithm is written simply as ________.

Part B: Solve the Following

1. Find (log10).
2. Find (log10000).
3. Find (log0.001).
4. Evaluate (log(100 * 1000)).
5. Evaluate (log(1000 / 100)).

Part C: Short Questions

1. Define the common logarithm.
2. What is the base of the common logarithm?
3. Write the common logarithm formula.
4. State the product rule of logarithms.
5. List any three applications of common logarithms.

Part D: Thinking Questions

1. Explain why (log1 = 0).
2. Why are logarithms useful in science and engineering?
3. How does a common logarithm simplify multiplication?

You May be Interested In:

What is Logarithm of a Number? | Logarithm Meaning, Log 10, Natural Log & Log Rules

Conversion of Numbers from Ordinary Notation to Scientific Notation

Scientific Notation: Application of Real Numbers in Daily Life

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