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CAT Definition of logarithm - Practice Questions & MCQ

Edited By admin | Updated on Oct 04, 2023 04:20 PM | #CAT

Quick Facts

4 Questions around this concept.

Solve by difficulty

EASY

log_a p < log_a q , this expression is true when 1 < a and 0 < p < q. (True/False)

True

False

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Which of the following option is correct

$\log_45>\log_46$

$\log_{0.2}8<\log_{0.2}4$

$\log_{0.3}7>\log_{0.3}4$

All of the above.

View Solution

When 0 < a < 1 and m < n, then, log_a m __________ log_an?

greater than

less than

View Solution

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Concepts Covered - 1

Definition of logarithm

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 $log_{b}x$ and is defined as:

$\mathrm{b^y=x \Longrightarrow y=\log _b x}$

Where:

- b is the base of the logarithm.

- x is the number.

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

Note: $\mathrm{ b > 0 , (b\neq 1), and x > 0 .}$

Common Bases:

1. Natural Logarithm: Base e (where e $\approx$ 2.71828) . Denoted as $\mathrm{In \: x}$ .

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

Solved Examples:

1. If $\mathrm{3^{5}=243}$ , what is $\mathrm{log_3}$ 243 ?

Solution:

Using the definition of logarithm, we have:

$\mathrm{3^{y}=243}$

Given, $\mathrm{3^{5}=243}$

So, y = 5

Thus, $\mathrm{log_3 \: 243 = 5 .}$

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

Solution:

Using properties of logarithms:

$\mathrm{\begin{aligned} & \log 25=\log 5^2 \\ & 2 \log 5=1.3979 \\ & \log 5=\frac{1.3979}{2}=0.699 . \end{aligned}}$

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:

$\mathrm{\begin{aligned} -\log _b(m n) & =\log _b m+\log _b n \\ -\log _b\left(\frac{m}{n}\right) & =\log _b m-\log _b n \\ -\log _b\left(m^n\right) & =n \log _b m \end{aligned}}$

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:

$\mathrm{\log _b a=\frac{\log _c a}{\log _c b}}$

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.