📘 Definitions (Logarithms)
Learn and master Class 9 Mathematics Chapter 2 Definitions on Logarithms. This page includes free downloadable notes covering all essential definitions, formulas, and concepts as per the PECTAA 2025 syllabus.
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Chapter 2
Logarithms
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Sets and Functions
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Factorization and Algebraic Manipulation
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Trigonometry
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Chapter 2 Definitions - Logarithms
This section contains all essential definitions and concepts from Chapter 2: Logarithms.
Understanding the fundamental definitions is crucial for mastering logarithms. This collection covers all key terms, formulas, and concepts that form the foundation of logarithmic operations and their applications.
The definitions cover scientific notation, logarithm concepts, common and natural logarithms, characteristic and mantissa, and antilogarithms.
Key Definitions Covered
- Scientific Notation
- Logarithm of a Real Number
- Common Logarithm (Brigg's Logarithm)
- Natural Logarithm (Napier Logarithm)
- Characteristic and Mantissa
- Antilogarithm
- Difference between Common and Natural Logarithms
Study Tip:
Memorize these definitions thoroughly as they form the basis for solving all logarithmic problems. Understanding the relationship between logarithmic and exponential forms is particularly important.
Detailed Definitions
1
Scientific Notation
A number written in scientific notation is expressed as:
$a \times 10^n$, where $1 \leq a < 10$ and $n \in Z$
Here $a$ is called the coefficient or base number.
!
Note:
- If the number is greater than 1, then $n$ is positive.
- If the number is less than 1, then $n$ is negative.
2
Logarithm of a Real Number
The logarithm of a real number tells us how many times one number must be multiplied by itself to get another number.
The general form of a logarithm is: $\log_b x = y$ where:
- $b$ is the base
- $x$ is the result or the number whose logarithm is being taken
- $y$ is the exponent or the logarithm of $x$ to the base $b$
This means that $b^y = x$.
$\log_b x = y \Leftrightarrow b^y = x$ where $b > 0$, $x > 0$ and $b \neq 1$
3
Common Logarithm or Brigg's Logarithm
If the base of logarithm is taken as 10 then logarithm is called common logarithm or Brigg's logarithm.
It is written as $\log_{10} \_\_\_$ or simply as $\log\_\_$ (when no base is mentioned, it is usually assumed to be base 10).
4
Natural Logarithm
Logarithm having base $e$ is called Napier logarithm or Natural logarithm.
It is commonly written as $\ln(x)$ and is used extensively in higher mathematics, particularly in calculus and applications involving growth and decay processes.
5
Characteristic and Mantissa
The integral part of the logarithm of any number is called the characteristic and the decimal part of the logarithm of a number is called the mantissa and is always positive.
For example, if $\log 278.23 = 2.4443$ then:
- Characteristic is 2
- Mantissa is 0.4443
6
Antilog
The number whose logarithm is given is called antilogarithm.
If $\log y = x$, then $y$ is the antilogarithm of $x$, or $y = Anti \log x$
In other words, antilog is the inverse of a logarithm.
7
Difference between Common and Natural Logarithms
| Common Logarithm | Natural Logarithm |
|---|---|
| The base of a common logarithm is 10. | The base of a natural logarithm is $e$. |
| It is written as $\log_{10}(x)$ or simply $\log(x)$ when no base is specified. | It is written as $\ln(x)$. |
| Common logarithms are widely used in everyday calculations, especially in scientific and engineering applications. | Natural logarithms are commonly used in higher-level mathematics, particularly calculus and applications involving growth/decay processes. |
Important Note:
These definitions provide the foundation for understanding and working with logarithms. Make sure you understand each concept thoroughly before moving on to exercises and applications.
Created by Hira Science Academy | Aligned with PECTAA 2025 Syllabus