CAT Properties of logarithm - Practice Questions & MCQ

Logarithms, being the inverse operations to exponentiation, have a set of properties that arise from the properties of powers. Understanding these properties is crucial for simplifying and solving logarithmic expressions and equations.

Properties of Logarithms:

1. Product Rule:  

2. Quotient Rule:  

3. Power Rule:  

4. Change of Base Formula:  

Where  c can be any positive number different from 1.

5. Identity:  

6. Zero Property:  

Solved Example:

1. Simplify:

Solution:  

   Using the quotient rule:

2. If  log 5 = 0.69897 , find the value of log 25 .

Solution:  

   Using the power rule:

Tips and Tricks:

1. Memorise the Properties: While understanding the derivation of each property is beneficial, ensuring that you've memorised them will speed up problem-solving.

2. Practice Simplification: Many questions involve simplifying complex logarithmic expressions using the properties. Regular practice will make this process intuitive.

3. Stay Aware of Valid Domains: Remember that the value inside a logarithm (called the argument) should always be positive. Also, the base should be positive and not equal to 1.

4. Use the Change of Base Formula Sparingly: While it's an essential tool, often problems can be solved without changing the base if you're familiar enough with the other properties.

5. Link with Exponents: Since logarithms are the inverse of exponentiation, think in terms of powers when stuck. This can sometimes provide a fresh perspective on a challenging problem.

In summary, the properties of logarithms are powerful tools that allow for the simplification and solution of logarithmic problems. They form the backbone of many quantitative questions in management entrance exams. With a strong grasp of these properties and regular practice, students will be well-prepared to tackle any logarithmic challenge.