CAT Definition of logarithm - Practice Questions & MCQ

A logarithm is a mathematical operation that determines how many times one number should be multiplied by itself to obtain another number. It's the inverse operation to exponentiation.

Definition:

The logarithm of a number x with base b is denoted as and is defined as:

Where:

- b is the base of the logarithm.

- x is the number.

- y is the logarithm of x to the base b .

Note:

Common Bases:

1. Natural Logarithm: Base e (where e 2.71828) . Denoted as .

2. Common Logarithm: Base 10. Denoted as log x.

Solved Examples:

1. If , what is   243 ?

Solution:  

   Using the definition of logarithm, we have:

   Given,

   So, y = 5  

   Thus,

2. Given log 25 = 1.3979 , find  log 5.

Solution:  

   Using properties of logarithms:

Tips and Tricks:

1. Understand the Basics: Ensure you're clear on the foundational concept that logarithm is the inverse operation of exponentiation.

2. Use Properties of Logarithms: Familiarise yourself with various properties like:

3. Change of Base Formula: If you're unfamiliar with the base used in a question, remember you can change the base using the formula:

   Where c can be any positive number different from 1.

4. Stay Cautious: Remember that the logarithm of a number to a base less than 1 is negative. Also, the base and the number whose logarithm we are taking should always be positive.

5. Practice with Calculator: While understanding the concepts is crucial, proficiency with a calculator can speed up solving problems, especially when dealing with non-integral logarithm values.

In summary, understanding the definition and properties of logarithms is fundamental for solving a variety of quantitative problems in management entrance exams. With consistent practice and a clear grasp of the basics, students can confidently tackle logarithmic questions.