## Introduction

In the CBSE board, chapters of Logarithm are included in the syllabus of classes 9, 10, and 11. Students of class 9 will be introduced to Logarithm questions and answers for the very first time. Hence, the thorough practice of Logarithm problems and answers is the need of the hour.

However, before proceeding with the chapter on Logarithm, students should be absolutely clear on the basic concepts. It is only then that solving difficult Logarithm questions would become considerably easier.

### Questions Based on Logarithm

Here are some of the Logarithm questions that would impart some ideas to students.

Question 1: Find Out the Incorrect Statement from Below -

(a) log (1 + 2 + 3) = log 1 + log 2 + log 3

(b) log (2 + 3) = log (2 x 3)

(c) log10 10 = 1

(d) log10 1 = 0

Solution: The answer is option (b) log (2 + 3) = log (2 x 3).

Question 2: What is the Value of log5512, when log 2 = 0.3010 and log 3 = 0.4771?

(a) 3.912

(b) 3.876

(c) 2.967

(d) 2.870

Solution: The answer is option (b) 3.876.

Question 3: Find the Value of log 9, When log 27 Amounts to 1.431.

(a) 0.954

(b) 0.945

(c) 0.958

(d) 0.934

Solution: An answer is an option (a) 0.954.

Question 4: What is the Value of log2 10, When log10 2 = 0.3010?

(a) 1000/301

(b) 699/301

(c) 0.6990

(d) 0.3010

Solution: An answer is an option (a) 1000/301.

Question 5: What is the Value of log10 80, When log10 2 = 0.3010?

(a) 3.9030

(b) 1.9030

(c) 1.6020

(d) None of the above option

Solution: An answer is an option (b) 1.9030.

Question 6: How Many Digits are there in 264 When log 2 = 0.30103?

(a) 21

(b) 20

(c) 18

(d) 19

Solution: The answer is option (b) 20.

Question 7: Which of the Following is True, if ax = by?

(a) log a/log b = x/y

(b) log a/b = x/y

(c) log a/log b = y/x

(d) None of the above option

Solution: The answer is option (c) log a/log b = y/x.

Question 8: What is the Value of log2 16?

(a) 8

(b) 4

(c) 1/8

(d) 16

Solution: The answer is (b) 4.

Question 9: Find the Value of y, if logx y = 100 and log2 x = 10.

(a) 21000

(b) 210

(c) 2100

(d) 210000

Solution: The answer is option (a) 21000.

Question 10: Find the Value of log10 (0.0001).

(a) – 1/4

(b) 1/4

(c) 4

(d) - 4

Solution: The answer is option (d) – 4.

Question 11: What is the Value of x When log2 [log3 (log2x)] = 1?

(a) 512

(b) 12

(c) 0

(d) 128

Solution: The answer is an option (a) 512.

Students’ queries simply and straightforward on Logarithm questions can be clarified in Vedantu’s online classes. You also have the option of downloading PDF materials from the official website. Download the app today!

Logarithm problems and answers are offered for students to solve to fully grasp the subject. These questions are based on the Logarithm chapter in the syllabus for Classes 9, 10, and 11. Practicing these problems will not only help students perform well in academic examinations, but will also allow them to compete in state or national level competitions such as Maths Olympiad.

The Logarithmic function is the exponential function's inverse. It is defined as follows:

If and only if x=ay, then y=logax; for x>0, a>0, and a1.

Logarithms are another way of expressing exponents in Mathematics. A number's Logarithm with a base equals another number. Exponentiation is the inverse function of Logarithm. For instance, if 102 = 100, log10 100 = 2.

As a result, we may deduce that

bn = x or log b x = n

Where b is the Logarithmic function's base.

This may be translated as "the Logarithm of x to the base b equals n."

In this post, we will cover the concept of Logarithms, the two types of Logarithms (common Logarithm and Natural Logarithm), and several Logarithmic characteristics with numerous solved cases.

### History

In the 17th century, John Napier invented the notion of Logarithms. Later, it was utilized by many scientists, navigators, engineers, and others to accomplish numerous computations, which simplified it. Logarithms, in a nutshell, are the inverse process of exponentiation.

### What Exactly are Logarithms?

A Logarithm is defined as the number of powers to which a number must be increased to obtain some other numbers. It is the simplest way to express enormous numbers. A Logarithm has many key features that demonstrate that Logarithm multiplication and division may also be stated in the form of Logarithm addition and subtraction.

"The exponent by which b must be raised to give an is the Logarithm of a positive real number a concerning a base b, a positive real number not equal to 1[nb 1]."

in other words, by=a ⇔logba=y

Where,

The letters "a" and "b" represent two positive real numbers.

y is a valid number.

"a" is the argument, which is located within the log, and "b" is the base, which is located at the bottom of the log.

In other terms, the Logarithm answers the question, "How many times is a number multiplied to get the other number?"

For example, how many threes must be multiplied to reach the answer 27?

When we multiply 3 by 3, we get the answer 27.

As a result, the Logarithm is 3.

The Logarithmic form is as follows:

Log3 (27) = 3..... (1)

As a result, the base 3 Logarithm of 27 is three.

The Logarithm form shown above can alternatively be expressed as:

3x3x3 = 27

33 = 27..... (2)

As a result, equations (1) and (2) have the same meaning.

### Types of Logarithms

In most circumstances, we are dealing with two forms of Logarithms, notably

Common Logarithms

Natural Logarithms

### Common Logarithms

The base 10 Logarithms are another name for the common Logarithm. It is denoted by log10 or just log. The common Logarithm of 1000, for example, is expressed as a log (1000). The common Logarithm specifies how many times the number 10 must be multiplied to obtain the appropriate result.

For instance, log (100) = 2

When we multiply 10 by itself twice, we obtain the number 100.

### Natural Logarithm

The base e Logarithm is another name for the natural Logarithm. The natural Logarithm is denoted by the symbols ln or loge. In this case, "e" stands for Euler's constant, which is approximately equal to 2.71828. The natural Logarithm of 78, for example, is represented as ln 78. The natural Logarithm specifies how many times "e" must be multiplied to obtain the appropriate result.

For instance, ln (78) = 4.357.

As a result, the base e Logarithm of 78 is 4.357.

### Logarithm Properties and Rules

Logarithmic operations can be conducted according to specific rules. These rules are known as:

Product rule

Division rule

Power rule/Exponential Rule

Change of base rule

Base switch rule

Derivative of log

Integral of log

## FAQs on Logarithm Questions

1. How to Solve Logarithm Basic Questions?

While the solution to be used will vary among logarithmic functions questions, the basic steps involve the following –

Determining the number of problems present in the logarithm

Apply relevant properties for simplification of the problem

the problem has to be rewritten sans logarithms

simplify problem further

find the solution of x, and

check the final solution. It must be noted for logs questions that logarithm of a negative number cannot be taken.

2. What are the Different Properties to Keep in Mind for Solving Log Maths Questions?

There are four properties to be followed for solving logarithm questions. The properties are – (1) product property, (2) quotient property, (3) power property, and (4) change of base property.

Product rule indicates that multiplying two or more logarithms with common bases becomes equal to the value arising out of separate logarithms. Quotient property lays down that two logarithms having same bases amounts to be equal to result generated from the difference in logarithms. Moreover, in case of change of base property, a given logarithm can be written with a new base.

3. What are Logarithmic Functions?

The inverse of exponential functions is termed as logarithmic functions. The logarithmic function, y = log_{a}x corresponds to the exponential equation, x = ay. y = log_{a}x. However, this relation holds only under a specific condition. Only if – (1) x = a^{y}, (2) a > 0, and (3) a≠1, will the relation be applicable.

Having a clear idea about logarithmic functions is essential for solving even the basic log questions.

4. Where can I get good study materials for Logarithms?

Everything you need may be found on the Vedantu app or website. These materials are prepared by experts in the subject, and the information is accurate and dependable. Students will be able to obtain revision notes, important questions, question papers, and much more! There are no fees or costs associated with these study resources. All students need to do is sign in, and then they can download everything they want in pdf format. You may benefit from these free tools, which will undoubtedly help you ace your tests.

5. Is it required to complete all of the exercises provided?

It is always preferable to complete all of the chapter activities because they provide practice. Maths is not a topic that can be learned simply by reading or memorizing. It takes dedication and practice. After each part of the NCERT textbook, there are examples and activities. After you've finished each lesson, check through the solved examples and make sure you understand them properly. Then go to the exercises and attempt to solve them. There is a potential that identical problems will appear in the test, therefore students should practice the sums in the exercises thoroughly.

6. How can I perform well in Math with the help of Vedantu?

The best way to do well in class 9 Maths is to practice. Complete all of the chapter exercises. This will enhance your problem-solving abilities as well as your speed and efficiency. Important formulae, definitions, and equations can be written down in a notebook and reviewed regularly. Solve previous year's sample papers and question papers within a time restriction, all of these resources are available for free on Vedantu. This will familiarise you with the paper design and question style, as well as help you improve your time management abilities. Revise frequently to ensure that you retain all you've learned for a longer amount of time.

7. What exactly are Logarithms and their two types?

Exponents can also be expressed using Logarithms. A Logarithm is defined as the number of powers to which a number must be increased to obtain some other numbers. In other terms, it answers the question, "How many times is one number multiplied to produce the other number?" A number's Logarithm is written as logb x = y. The following are the two most prevalent types of Logarithms:

Base 10 Logarithm (or Common Logarithm)

Base e Logarithm (or Natural Logarithm)