Logarithm
Table of Contents
- Introduction
- History
- Relation Between Exponents & Logarithms
- Definition
- Important Terms
- Laws of Logarithm
- Change of Base Formula
- Characteristic & Mantissa
- Common & Natural Logarithm
- Antilogarithm
- Solved Problems
- Graph of Logarithm
- Applications
- Important Formula List
- Practice Questions
- Long Questions with Answers
- Summary Notes
1. Introduction to Logarithm
Logarithm is a mathematical method used to simplify multiplication, division, powers and roots.
Logarithms are closely related to exponents.
Logarithmic form:
This means: The power of 10 required to get 100 is 2.
2. History of Logarithm
Logarithms were invented by Scottish mathematician John Napier.
They helped scientists perform difficult calculations before calculators were invented.
3. Relation Between Exponents and Logarithms
Exponential Form:
Logarithmic Form:
Example
4. Definition of Logarithm
If:
then:
Conditions:
- a > 0
- a ≠ 1
- N > 0
5. Important Terms
Base
Base = 2
Argument
Argument = 8
Logarithm
Logarithm = 3
6. Laws of Logarithm
1. Product Rule
2. Quotient Rule
3. Power Rule
4. Log of 1
5. Log of Base
7. Change of Base Formula
Example
8. Characteristic and Mantissa
Characteristic = 2
Mantissa = 0.5529
Rule for Numbers Greater Than 1
Characteristic is one less than number of digits before decimal point.
Rule for Numbers Less Than 1
9. Common and Natural Logarithm
Common Logarithm
Base = 10
Natural Logarithm
Base = e = 2.71828...
10. Antilogarithm
Antilogarithm is the inverse process of logarithm.
Then:
11. Solved Problems
Example 1
Example 2
Example 3
12. Graph of Logarithmic Function
- Defined only for x > 0
- Passes through (1,0)
- Slowly increases
- Never touches y-axis
13. Applications of Logarithm
- Earthquake measurement
- Sound measurement
- pH value calculation
- Banking and compound interest
- Computer science
- Astronomy
14. Important Formula List
| Formula | Meaning |
|---|---|
| logₐ1 = 0 | Log of 1 |
| logₐa = 1 | Log of base |
| logₐ(MN) | Product Rule |
| logₐ(M/N) | Quotient Rule |
| logₐ(Mⁿ) | Power Rule |
15. Practice Questions
Very Short Questions
- Find log₂16
- Find log₁₀1000
- What is logₐ1?
- What is logₐa?
Short Questions
- Solve log₅x = 3
- Simplify log 4 + log 25
- Simplify log 100 − log 10
16. Long Questions with Answers
1. Prove Product Law of Logarithm
Let:
Hence proved.
2. Explain Characteristic and Mantissa
Characteristic = 2
Mantissa = 0.5529
Characteristic is integer part and mantissa is decimal part.
3. Derive Change of Base Formula
Hence proved.
17. Summary Notes
- Logarithm is inverse of exponent.
- Product becomes addition.
- Division becomes subtraction.
- Power comes in front.
- Common logarithm has base 10.
- Natural logarithm has base e.