Chapters

1 Real Numbers 2 Logarithms 3 Sets and Functions 4 Factorization 5 Linear Equations 6 Trigonometry 7 Coordinate Geometry 8 Logic 9 Similar Figures 10 Graphs of Functions 11 Loci and Construction 12 Information Handling 13 Probability

Unit 2: Logarithms

Class 9 Mathematics | Punjab Curriculum and Textbook Board Syllabus 2025

1. Define scientific notation.

A number written in scientific notation written as:

\( a \times 10^{n},\ where\ 1 \leq a < 10\ and\ n \in Z \)

Here \( a \) is called the coefficient or base number.

2. What is the 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, and
  • \( y \) is the exponent or the logarithm of \( x \) to the base \( b \).

This means that \( b^{y} = x \).

OR

The logarithm of \( x \) to the base \( b \) is \( y \), means that when \( b \) is raised to the power \( y \), it equals \( x \). The relationship between logarithmic form and exponential form is given below:

\( \log_{b}x = y \Longleftrightarrow b^{y} = x\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ where\ b > 0,\ x > 0\ and\ b \neq 1 \)

3. Define 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. Define Natural logarithm.

Logarithm having base \( e \) is called Napier logarithm or Natural logarithm.

5. Differentiate between 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 \( 2 \) and mantissa is \( 0.4443 \)

6. Define antilog.

The number whose logarithm is given is called antilogarithm. \( i.e. \) 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. What is the difference between Common and Natural Logarithms?

Common Logarithm Natural Logarithm
The base of a common logarithm is \( \mathbf{10} \). The base of a natural logarithm is \( e \).
It is written as \( \mathbf{\log}_{\mathbf{10}}\left( \mathbf{x} \right) \) or simply \( \mathbf{\log}\left( \mathbf{x} \right) \) 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.