Open App

UPSC Exam > UPSC Notes > CSAT Preparation > Important Concepts & Formulas: Logarithms

Important Logarithms Formulas for JEE and NEET

Table of contents
What are Logarithms?
Types of Logarithms
Logarithm Rules
Logarithm Properties
Characteristics and Mantissa
Important Conversions
Important Formulas
Solved Examples

What are Logarithms?

A logarithm is defined as the power to which a number must be raised to get some other values. It is the most convenient way to express large numbers. A logarithm has various important properties that prove multiplication and division of logarithms can also be written in the form of a logarithm of addition and subtraction.
“The logarithm of a positive real number a with respect to base b, a positive real number not equal to 1^{[nb 1]}, is the exponent by which b must be raised to yield a”.
i.e. b^y= a ⇔log_ba=y
Where,

“a” and “b” are two positive real numbers
y is a real number
“a” is called argument, which is inside the log
“b” is called the base, which is at the bottom of the log.

In other words, the logarithm gives the answer to the question “How many times a number is multiplied to get the other number?”.
For example, how many 3’s are multiplied to get the answer 27?
If we multiply 3 for 3 times, we get the answer 27.
Therefore, the logarithm is 3.
The logarithm form is written as follows:
Log₃ (27) = 3 ….(1)
Therefore, the base 3 logarithm of 27 is 3.
The above logarithm form can also be written as:
3 x 3 x 3 = 27
3³ = 27 …..(2)
Thus, the equations (1) and (2) both represent the same meaning.
Below are some of the examples of conversion from exponential forms to logarithms.
Important Logarithms Formulas for JEE and NEET

Question for Important Concepts & Formulas: Logarithms

Try yourself:What is the logarithm of 1000 with respect to base 10?

A.
log10(1000) = 2
B.
log10(1000) = 3
C.
log10(1000) = 4
D.
log10(1000) = 5

View Solution

Types of Logarithms

(i) Natural Logarithm: log_e N is called natural Logarithm or Naperian Logarithm denoted by (ln N) i.e., when logarithm’s base is “e” then it is called natural logarithm. Ex: log_e 7
(ii) Common Logarithm: log₁₀ N is called Brigg’s Logarithm when the base is 10. Ex: log₁₀100

Logarithm Rules

There are certain rules based on which logarithmic operations can be performed. The names of these rules are:

Product rule
Division rule
Power rule/Exponential Rule
Change of base rule
Base switch rule
Derivative of log
Integral of log

Let us have a look at each of these properties one by one

1. Product Rule

In this rule, the multiplication of two logarithmic values is equal to the addition of their individual logarithms.
Logb (mn)= logb m + logb n
Example: log3 ( 2y ) = log3 (2) + log3 (y)

2. Division Rule

The division of two logarithmic values is equal to the difference of each logarithm.
Log_b (m/n)= log_b m – log_b n
Example: log₃ (2/ y) = log₃ (2) -log₃ (y)

3. Exponential Rule

In the exponential rule, the logarithm of m with a rational exponent is equal to the exponent times its logarithm.
Log_b (mⁿ) = n log_b m

Example: log_b(2³) = 3 log_b 2

4. Change of Base Rule

Log_b m = log_a m/ log_a b
Example: logb 2 = loga 2/loga b

5. Base Switch Rule

logb (a) = 1 / loga (b)
Example: log_b 8 = 1/log₈ b

6. Derivative of log

If f (x) = log_b (x), then the derivative of f(x) is given by;
f'(x) = 1/(x ln(b))

Example: Given, f (x) = log₁₀ (x)
Then, f'(x) = 1/(x ln(10))

7. Integral of Log

∫log_b(x)dx = x(log_b(x) – 1/ln(b)) + C

Example: ∫ log₁₀(x) dx = x ∙ (log₁₀(x) – 1 / ln(10)) + C

Logarithm Properties

log_a 1 = 0, a > 0, a $\neq$ 1
log_a a = 1, a > 0, a $\neq$ 1
Log_a a^x = x, x $\forall$ x $\in$ R, x >0
a^{log_a x} = x, x $\forall$ x $\in$ R, x >0
log_a (m.n) = log_a m + log_a n, n $\forall$ m, n > 0, a > 0, a $\neq$ 1
log_a (m/n) = log_a m – log_a n, n $\forall$ m, n > 0, a > 0, a $\neq$ 1
log_a (mⁿ) = n log_a m, m $\forall$ m, m > 0, a > 0, a $\neq$ 1
log_a (1/m) = – log_a m, m $\forall$ m, n > 0, a > 0, a $\neq$ 1
log_a b = x $\forall$ a, b > 0, a $\neq$ 1 and x $\in$ R
(i) log_1/a b = – x
(ii) log_a (1/b) = – x
(iii) log_1/a (1/b) = -x
log_a^m b = 1/m log_a b
log_a x is a decreasing function, if 0 < a < 1
log_a x is a increasing function, if a > 1
When 0 < a < 1 then
- log_a b $\geq$ log_a c, $\Leftrightarrow$ b $\leq$ c
- log_a b $\geq$ c, $\Leftrightarrow$ b $\leq$ a^c
When a >1
- log_a b $\geq$ log_a c, $\Leftrightarrow$ b $\geq$ c > 0
- log_a b $\geq$ c, $\Leftrightarrow$ b $\geq$ a^c

Question for Important Concepts & Formulas: Logarithms

Try yourself:Which type of logarithm is denoted by ln N when the base is "e"?

A.
Natural logarithm
B.
Common logarithm
C.
Exponential logarithm
D.
None of these

View Solution

Characteristics and Mantissa

Characteristic: The integral part of logarithm is known as characteristic.
Mantissa: The decimal part is known as mantissa and is always positive

Important Logarithms Formulas for JEE and NEET In log 3274 = 3.5150, the integral part is 3 i.e., characteristic is 3 and the decimal part is .5150 i.e., mantissa is .5150.

Points to Remember About Characteristics

The characteristic of common logarithm of positive number less than unity (i.e.,1) is negative.
The characteristic of common logarithm of a positive number greater than 1 is positive.
If the logarithm to any base a gives the characteristic ‘n’, then the number of possible integral values is given by aⁿ⁺¹ −aⁿ. For example log₁₀ x = n.abcd, then the number of integral values that x can have given by 10ⁿ⁺¹ −10ⁿ
If the characteristic of log₁₀ x is negative (i.e., − n), then the number of zeros between the decimal and the first significant number after the decimal is (n −1)

Important Conversions

For a > 1, a 1, c > 0, a ^b > c $\Leftrightarrow$ log_a c < b
For 0 < a < 1, a 1, c > 0, a ^b > c $\Leftrightarrow$ log_a c > b

Important Formulas

Important Logarithms Formulas for JEE and NEET

log_b(mn) = log_b(m) + log_b(n)
log_b(m/n) = log_b (m) – log_b(n)
Log_b (xy) = y log_b(x)
Log_bm√n = log_b n/m
m log_b(x) + n log_b(y) = log_b(x^myⁿ)
log_b(m+n) = log_b m + log_b(1+nm)
log_b(m – n) = log_b m + log_b (1-n/m)

Solved Examples

Question 1: If log₂X + log₄X = log_0.25√6 and x > 0, then x is:

A. 6^-1/6

B. 6^1/6

C. 3^-1/3

D. 6^1/3

Correct Answer is Option (A).

log₂x + log₄x = log_0.25√6
We can rewrite the equation as:

⇒ log₂x * 3 = 2log_0.25√6
⇒ log₂x³ = -log₄6
⇒
⇒
⇒ 2log₂x³ = -log₂6
2log₂x³ + log₂6 = 0
log₂6X⁶ = 0
6x⁶ = 1
x⁶ = 1/6
The question is "If log₂X + log₄X = log_0.25 √6 and x > 0, then x is"
Hence, the answer is "6^-1/6".

Question 2: log₉ (3log₂ (1 + log₃ (1 + 2log₂x))) = 1/2. Find x.

A. 4

B. 1/2

C. 1

D. 2

Correct Answer is Option (D).

log₉ (3log₂ (1 + log₃ (1 + 2log₂x)) = 1/2

3log₂(1 + log₃(1 + 2log₂x)) = 9^1/2 = 3
log₂(1 + log₃(1 + 2log₂x) = 1
1 + log₃(1 + 2log₂x) = 2
log₃(1 + 2log₂x) = 1
1 + 2log₂x = 3
2log₂x = 2
log₂x = 1
x = 2

The question is "Find x."

Hence, the answer is "2".

Question 3: If 2^2x+4 – 17 × 2^x+1 = –4, then which of the following is true?

A. x is a positive value

B. x is a negative value

C. x can be either a positive value or a negative value

D. None of these

Correct Answer is Option (C).

2^x+4 – 17 * 2^x+1 = – 4
=> 2^x+1 = y
2^2x+2 = y²
2²(2^2x+2) – 17 * 2^x+1 = –4
4y² – 17y + 4 = 0
4y² – 16y – y + y = 0
4y (y – 4) – 1 (y – 4) = 0

y = 1/4 or 4

2^x+1 = 1/4 or 4

⇒ x + 1 = 2 or – 2
x = 1 or – 3

The question is "which of the following is true?"

Hence, the answer is "x can be either a positive value or a negative value".

Question 4: If log₁₂27 = a, log₉16 = b, find log₈108

A. Important Logarithms Formulas for JEE and NEET

B. Important Logarithms Formulas for JEE and NEET

C. Important Logarithms Formulas for JEE and NEET

D. Important Logarithms Formulas for JEE and NEET

Correct Answer is Option (D).

log₈108 = log₈(4 * 27)
log₈108 = log₈4 + log₈27
⇒ log₈4 = 2/3

Important Logarithms Formulas for JEE and NEET

log₈27 = 2 * log₁₆9
log₉16 = b
log₁₆9 = 1/b

log₈27 = 2/b

Important Logarithms Formulas for JEE and NEET

The question is "find log₈108."

Hence, the answer is

Question 5: Important Logarithms Formulas for JEE and NEET If a, b are integers such that x = a, and x = b satisfy this inequation, find the maximum possible value of a – b.

A. 214

B. 216

C. 200

D. 203

Correct Answer is Option (A).

Important Logarithms Formulas for JEE and NEET

log₃x = y

⇒ Important Logarithms Formulas for JEE and NEET

y ∈ (3, 5)
3 < log₃x < 5
27 < x < 243
Therefore max ( a – b) will be when a = 242 and b = 28. Therefore, max(a – b) = 214.

The question is "find the maximum possible value of a – b."

Hence, the answer is "214".

Question 6: log₅x = a (This should be read as log X to the base 5 equals a) log₂₀x = b. What is log_x10?

A. Important Logarithms Formulas for JEE and NEET
B. (a + b) * 2ab

C. Important Logarithms Formulas for JEE and NEET
D.

Correct Answer is Option (A).

Given, log₅x = a
log₂₀x = b
log_x5 = 1/a

log_x20 = 1/b
Important Logarithms Formulas for JEE and NEET

⇒

The question is "What is log_x10?"

Hence, the answer is

Question 7: log₃x + log_x3 = 17/4. Find x.

A. 34

B. 3^1/8

C. 3^1/4

D. 3^1/3

Correct Answer is Option (C).

log₃x + log_x3 = 17/4

Let y = log₃x
We know that log_x3 = Important Logarithms Formulas for JEE and NEET

Hence log_x3 = 1/y

Thus the equation can be written as Important Logarithms Formulas for JEE and NEET

4y² + 4 = 17y
4y² + 4 - 17y = 0
Solving the above equation we get y = 4 or 1/4

If y = 4
log₃x = 4
Then x = 3⁴
If y = 1/4

log₃x = 1/4

Then x = 3^1/4
The question is "Find x."

Hence, the answer is "3⁴".

Question 8: log_xy + log_yx² = 3. Find log_xy³.

A. 4

B. 3

C. 3^1/2

D. 3^1/16

Correct Answer is Option (B).

log_xy + log_yx² = 3
Let a = log_xy

Important Logarithms Formulas for JEE and NEET

log_yx² = 2log_yx
We know that log_yx = Important Logarithms Formulas for JEE and NEET

Hence form above log_yx = 1/a

Now rewritting the equation log_xy + log_yx² = 3

Using a we get Important Logarithms Formulas for JEE and NEET

i.e., a² - 3a + 2 = 0
Solving we get a = 2 or 1
If a = 2, Then log_xy = 2 and log_yx³ = 3
log_xy = 3 * 2 = 6
Or
If a = 1, Then log_xy = 1 and log_yx³ = 3
log_xy = 3 * 1 = 3

The question is "Find log_xy³."

Hence, the answer is "3".

Question 9: log₂ 4 * log₄ 8 * log₈ 16 * ……………nth term = 49, what is the value of n?

A. 49

B. 48

C. 34

D. 24

Correct Answer is Option (B).

First, the nth term of L.H.S need to be defined by observing the pattern :-
It is log_(2ⁿ) 2.2ⁿ
Given,
log₂ 4 * log₄ 8 * log₈ 16 * ……………log_(2ⁿ) 2.2n = 49
Whenever solving a logarithm equation, generally one should approach towards making the base same.
Making the base 2:-

Important Logarithms Formulas for JEE and NEET

log_(2ⁿ) 2.2ⁿ = 49
log_(2ⁿ) 2 + log_(2ⁿ) 2ⁿ = 49
1 + n = 49
n = 48

The question is "what is the value of n?"

Hence, the answer is "48".

Question 10: If 3^{3 + 6 + 9 + ……… 3x}= Important Logarithms Formulas for JEE and NEET what is the value of x?

A. 3

B. 6

C. 7

D. 11

Correct Answer is Option (D).

First of all, let us define the x^th term.

Important Logarithms Formulas for JEE and NEET

Whenever you encounter a distinctive number such as one given in R.H.S of above equation, always try to find its significance in the context of question.

In this case L.H.S has 3^a, so Important Logarithms Formulas for JEE and NEET must be some form of 3^a.

With little hit and trial, you may find Important Logarithms Formulas for JEE and NEET

3^{3(1 + 2 + 3 + ...X)} = 3^{-3 * -66}
⇒ 3³ * 3^x(x+1)/2 = 3^3*66

Important Logarithms Formulas for JEE and NEET

x(x+1) = 132
Solving this equation for x > 0, we get x = 11.
You should directly be able to see that 132 = 11 * 12 => x= 11
And avoid wasting time solving the complete equation.

The question is "what is the value of x?"

Hence, the answer is "11".

Question 11: x, y, z are 3 integers in a geometric sequence such that y - x is a perfect cube. Given, log₃₆x² + log₆√y + log₂₁₆y^1/2z = 6. Find the value of x + y + z.

A. 189

B. 190

C. 199

D. 201

Correct Answer is Option (A).

Let us begin with simplifying the equation:-
⇒ log_6²x² + log₆y^1/2 + 3log_6³y^1/2z = 6

Important Logarithms Formulas for JEE and NEET

log₆x + log₆y^1/2y^1/2z = 6
log₆xyz = 6
xyz = 6⁶

Given x,y,z is in G.P. Let x = a, y = ab, z = ab²
⇒ xyz = a³b³ = (ab)³
(ab)³ = (6²)³
Possible values of (a,b) satisfying the equation :-
(1, 36), (2, 18), (3, 12), (4, 9), (9, 4), (12, 3), (18, 2), (36, 1)
Given y-x is a perfect cube
⇒ ab-a is perfect cube
⇒ a(b-1) is perfect cube
Only possible when (a, b) = (9, 4)
∴ x = 9 , y = 36 , z = 144
∴ x + y + z = 9 + 36 + 144 = 189

The question is "Find the value of x + y + z."

Hence, the answer is "189".

Question 12: 10^{log(3 - 10logy)} = log₂(9 - 2^y), Solve for y.

A. 0

B. 3

C. 0 and 3

D. none of these

Correct Answer is Option (D).

Before beginning to simplify the equation, don’t forget that anything inside a log cannot be negative
10^log(3-y) = log₂(9 - 2^y) (y > 0)…………………………………(1)
3 - y = log₂(9 - 2^y) (Therefore, 3 - y > 0 =) (y < 3)) ……………………… (2)
2^3-y = 9 - 2^y
2^y = t

Important Logarithms Formulas for JEE and NEET

⇒ 8 = 9t –t²
⇒ t² - 9t + 8 = 0
⇒ t² - t - 8t - 8 = 0
⇒ t(t - 1) - 8(t - 1) = 0
⇒ t = 1, 8
Therefore, 2^y = 1 and 2^y = 8
⇒ y = 0 and y = 3
However, from inequalities (1) and (2), y cannot take either of these value.

The question is "Solve for y."

Hence, the answer is "none of these".

Question 13: 4^{6+12+18+24+…+6x} = (0.0625)^-84, what is the value of x?

A. 7

B. 6

C. 9

D. 12

Correct Answer is Option (A).

Take right side expression,

Important Logarithms Formulas for JEE and NEET

= (4^-2)^-84
= 4¹⁶⁸
Take left side expression
4^{6+12+18+24+…+6x} = 4^6(1+2+3+4+x)
= 4⁶ * 4^{(1+2+3+4+…+x)}
= 4⁶ * 4^x(x+1)/2 (using the formula for sum of natural numbers from 1 to x)
Equating left and right side expresssions, we get 4⁶ * 4^x(x+1)/2 = 4¹⁶⁸
Or 4⁶ * 4^x(x+1)/2 = 4^6*28

⇒ Important Logarithms Formulas for JEE and NEET

or x (x + 1) = 56
Solving for x we get, x = 7

The question is "what is the value of x?"

Hence, the answer is "7".

The document Important Logarithms Formulas for JEE and NEET is a part of the UPSC Course CSAT Preparation.

All you need of UPSC at this link: UPSC

	CSAT Preparation 218 videos\|139 docs\|151 tests

CSAT Preparation

218 videos|139 docs|151 tests

Join Course for Free

Top Courses for UPSC

View all

FAQs on Important Logarithms Formulas for JEE and NEET

1. What are the different types of logarithms?

Ans. There are two main types of logarithms: common logarithms, which have a base of 10, and natural logarithms, which have a base of e (Euler's number).

2. What are some important logarithm rules to remember?

Ans. Some important logarithm rules include the product rule (log(ab) = log(a) + log(b)), the quotient rule (log(a/b) = log(a) - log(b)), and the power rule (log(a^n) = n*log(a)).

3. What are some key logarithm properties to be aware of?

Ans. Some key logarithm properties include the fact that log(a) + log(b) = log(ab), log(a) - log(b) = log(a/b), and log(a^n) = n*log(a).

4. What are characteristics and mantissa in logarithms?

Ans. The characteristic in a logarithm is the integer part of the logarithm, while the mantissa is the decimal part. They together make up the complete logarithm value.

5. How can logarithms be used for important conversions and calculations?

Ans. Logarithms can be used to convert between exponential and logarithmic forms, solve exponential equations, and simplify complex mathematical calculations involving exponents.

Related Exams

UPSC Telangana Police Constable Quant CAT Super TET NDA SSC MTS / SSC GD Police Constable Exams Police SI Exams MAT

About this Document

	7.5K Views
	4.89/5 Rating
	Oct 24, 2024 Last updated

Document Description: Important Concepts & Formulas: Logarithms for UPSC 2024 is part of CSAT Preparation preparation. The notes and questions for Important Concepts & Formulas: Logarithms have been prepared according to the UPSC exam syllabus. Information about Important Concepts & Formulas: Logarithms covers topics like What are Logarithms?, Types of Logarithms, Logarithm Rules, Logarithm Properties, Characteristics and Mantissa, Important Conversions, Important Formulas, Solved Examples and Important Concepts & Formulas: Logarithms Example, for UPSC 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Important Concepts & Formulas: Logarithms.

Introduction of Important Concepts & Formulas: Logarithms in English is available as part of our CSAT Preparation for UPSC & Important Concepts & Formulas: Logarithms in Hindi for CSAT Preparation course. Download more important topics related with notes, lectures and mock test series for UPSC Exam by signing up for free. UPSC: Important Logarithms Formulas for JEE and NEET

Description

Full syllabus notes, lecture & questions for Important Logarithms Formulas for JEE and NEET - UPSC | Plus excerises question with solution to help you revise complete syllabus for CSAT Preparation | Best notes, free PDF download

Information about Important Concepts & Formulas: Logarithms

In this doc you can find the meaning of Important Concepts & Formulas: Logarithms defined & explained in the simplest way possible. Besides explaining types of Important Concepts & Formulas: Logarithms theory, EduRev gives you an ample number of questions to practice Important Concepts & Formulas: Logarithms tests, examples and also practice UPSC tests

	CSAT Preparation 218 videos\|139 docs\|151 tests

CSAT Preparation

218 videos|139 docs|151 tests

Join Course for Free

Download as PDF

Explore Courses for UPSC exam

Top Courses for UPSC

Explore Courses

Signup for Free!

Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.

Start learning for Free

10M+ students study on EduRev

Viva Questions

video lectures

past year papers

study material

Important Logarithms Formulas for JEE and NEET

ppt

Important Logarithms Formulas for JEE and NEET

Objective type Questions

Free

Summary

shortcuts and tricks

Extra Questions

Important questions

Semester Notes

Previous Year Questions with Solutions

Important Logarithms Formulas for JEE and NEET

practice quizzes

mock tests for examination

Exam

Sample Paper

MCQs

pdf

;

Additional Information about Important Concepts & Formulas: Logarithms for UPSC Preparation

Important Concepts & Formulas: Logarithms Free PDF Download

The Important Concepts & Formulas: Logarithms is an invaluable resource that delves deep into the core of the UPSC exam. These study notes are curated by experts and cover all the essential topics and concepts, making your preparation more efficient and effective. With the help of these notes, you can grasp complex subjects quickly, revise important points easily, and reinforce your understanding of key concepts. The study notes are presented in a concise and easy-to-understand manner, allowing you to optimize your learning process. Whether you're looking for best-recommended books, sample papers, study material, or toppers' notes, this PDF has got you covered. Download the Important Concepts & Formulas: Logarithms now and kickstart your journey towards success in the UPSC exam.

Importance of Important Concepts & Formulas: Logarithms

The importance of Important Concepts & Formulas: Logarithms cannot be overstated, especially for UPSC aspirants. This document holds the key to success in the UPSC exam. It offers a detailed understanding of the concept, providing invaluable insights into the topic. By knowing the concepts well in advance, students can plan their preparation effectively. Utilize this indispensable guide for a well-rounded preparation and achieve your desired results.

Important Concepts & Formulas: Logarithms Notes

Important Concepts & Formulas: Logarithms Notes offer in-depth insights into the specific topic to help you master it with ease. This comprehensive document covers all aspects related to Important Concepts & Formulas: Logarithms. It includes detailed information about the exam syllabus, recommended books, and study materials for a well-rounded preparation. Practice papers and question papers enable you to assess your progress effectively. Additionally, the paper analysis provides valuable tips for tackling the exam strategically. Access to Toppers' notes gives you an edge in understanding complex concepts. Whether you're a beginner or aiming for advanced proficiency, Important Concepts & Formulas: Logarithms Notes on EduRev are your ultimate resource for success.

Important Concepts & Formulas: Logarithms UPSC Questions

The "Important Concepts & Formulas: Logarithms UPSC Questions" guide is a valuable resource for all aspiring students preparing for the UPSC exam. It focuses on providing a wide range of practice questions to help students gauge their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation. The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level. Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.

Study Important Concepts & Formulas: Logarithms on the App

Students of UPSC can study Important Concepts & Formulas: Logarithms alongwith tests & analysis from the EduRev app, which will help them while preparing for their exam. Apart from the Important Concepts & Formulas: Logarithms, students can also utilize the EduRev App for other study materials such as previous year question papers, syllabus, important questions, etc. The EduRev App will make your learning easier as you can access it from anywhere you want. The content of Important Concepts & Formulas: Logarithms is prepared as per the latest UPSC syllabus.

Education Revolution

Signup to see your scores go up within 7 days!

Access 1000+ FREE Docs, Videos and Tests

Continue with Google

Takes less than 10 seconds to signup

Important Logarithms Formulas for JEE and NEET

What are Logarithms?

Types of Logarithms

Logarithm Rules

1. Product Rule

2. Division Rule

3. Exponential Rule

4. Change of Base Rule

5. Base Switch Rule

6. Derivative of log

7. Integral of Log

Logarithm Properties

Characteristics and Mantissa

Points to Remember About Characteristics

Important Conversions

Important Formulas

Solved Examples

Hence, the answer is "2".

Hence, the answer is "x can be either a positive value or a negative value".

Hence, the answer is

Hence, the answer is "214".

Hence, the answer is

Hence, the answer is "34".

Hence, the answer is "3".

Hence, the answer is "48".

Hence, the answer is "11".

Hence, the answer is "189".

Hence, the answer is "none of these".

Hence, the answer is "7".

CSAT Preparation

Top Courses for UPSC

FAQs on Important Logarithms Formulas for JEE and NEET

CSAT Preparation

Top Courses for UPSC

Viva Questions

video lectures

past year papers

study material

Important Logarithms Formulas for JEE and NEET

ppt

Important Logarithms Formulas for JEE and NEET

Objective type Questions

Free

Summary

shortcuts and tricks

Extra Questions

Important questions

Semester Notes

Previous Year Questions with Solutions

Important Logarithms Formulas for JEE and NEET

practice quizzes

mock tests for examination

Exam

Sample Paper

MCQs

pdf

Important Concepts & Formulas: Logarithms Free PDF Download

Importance of Important Concepts & Formulas: Logarithms

Important Concepts & Formulas: Logarithms Notes

Important Concepts & Formulas: Logarithms UPSC Questions

Study Important Concepts & Formulas: Logarithms on the App

Hence, the answer is "3⁴".