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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 CAT Course Quantitative Aptitude (Quant).

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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.

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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".

Quantitative Aptitude (Quant)

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FAQs on Important Logarithms Formulas for JEE and NEET

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Sample Paper

Important Logarithms Formulas for JEE and NEET

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Important Logarithms Formulas for JEE and NEET

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Hence, the answer is "3⁴".