Logarithm Formula: Types, Rules, Solved Examples

Content Curator

Logarithm formulas play a vital role in mathematics as well as physics as all the values in physics in exponential form to get accurate values final computation is only done by using logarithm values. These are extensively used in all fields of science, these also play a major role in surveying and navigation purposes.

Table of Content

What is a Logarithm?
Types of Logarithm
Logarithm Rules
Logarithm Formulas
Things to Remember
Previous Year Questions
Sample Questions

Keyterms: Log, Logarithm, Exponential functions, Integer, Base, Logarithm functions, Natural logarithmic functions, Common logarithmic functions

Also Read: Types of Probability

What is a Logarithm?

[Click Here for Sample Questions]

A logarithm is defined as the exponent or power up to which the base value should be raised. These are simply stated as,

a^m= n

Which states that m is the logarithm of n to base a.

Basic representation am= n can be written as m= logan

Example : 3³ = 27 or log₃ 27 = 3

Logarithm

Logarithm

Also Read:

Related Articles
Value of Log 0	Value of Log 1	Value of Log 1 to 10
Logarithm questions	Logarithmic Differentiation	Value of log infinity

Types of Logarithm

[Click Here for Previous Year Questions]

Logarithm is of two types namely,

Common Logarithm
Natural Logarithm

Common Logarithm

The common logarithm defines the number of times the value has to be multiplied by 10. It is simply called base 10 logarithms and can be represented as log or log10.

Example: log (100) = 2 or log 10 (100) = 2

Common Log	Natural Log
log₁₀x = log x	log_ex = In x

Natural Logarithm

Logarithms to the base e are called natural logarithms and are represented as ln n or loge where e is an Euler’s Constant and is approximately equal to 2.718.

Example: ln (78) or loge = 4.357

Also Read: Permutations and Combinations

Logarithm Rules

[Click Here for Sample Questions]

Let us look at the various rules in Logarithm.

Logarithm Product Rule

The logarithm of the product of two values is equal to the total sum of individual values.

It is represented as

logn(ab) = log_n a + log_n b

Logarithm Division Rule

Logarithm division of any two values is equal to the difference between the logarithm of individual values. It is represented as

log_n(a/b) = log_n a - log_n b

Exponent Rules	Logarithm Rules
\(a^m \times a^n = a^{m+n}\)	log(mn) = logm + logn
\(\frac{a^m}{a^n} = a^{m-n}\)	\(\log \frac{m}{n} = \log m - log n\)
(a^m)ⁿ= a^mn	log mⁿ = nlogm

Logarithm Power Rule

the logarithm of a positive value a to the power of b is equivalent to the product of b to the logarithm of a. It is represented as

log_n(ab) = b log_n(a)

Logarithm Zero Rule

The logarithm value of 1 is equivalent to zero. It is represented as

log_n (1) = 0

Logarithm Identity Rule

If the logarithm value and its base value are equal then it is equal to 0. It is represented as

log_n (n) = 1

Logarithm Exponential Rule

The logarithm of m with a rational exponent is equal to the exponent times its logarithm. It is represented as

Log_b (mⁿ) = n log_b m

Change of Base Rule

The logarithm of a value with a given base can be expressed in terms of the ratio with the same base that is different from the original base. It is represented as,

Log_b m = log_a m/ loga b

Also Read: Pascal’s Triangle

Logarithm Formulas

[Click Here for Previous Year Questions]

Here are some logarithm formulas with which we can solve a number of combinations of logarithms.

S.no	Formula
1	log_n (1) = 0
2	log_n (n) = 1
3	log_n (0) = -∞ if a>1 = ∞ if a<1
4	log_n(a/b) = log_n a - log_n b
5	log_n(ab) = log_n a + log_n b
6	log_n(ab) = b log_n(a)
7	log_nn√x = 1/n log_n(x)
8	log_n n = 1/log_n a
9	a log_n(x) = x
10	log_e(x) = ln x

Also Read:

Related Articles
Logarithm	Arithmetic Progression	Geometric Progression
Geometric Mean	Straight Lines	Angle Between Two Lines

Things to Remember

A logarithm is defined as the exponent or power up to which the base value should be raised.
Logarithms are used to make approximations for exact quantities
Small differences can be effectively calculated from these logarithms.
Chemical property can be determined effectively.
Logarithm is of two types namely common logarithm and natural logarithm.
The common logarithm defines the number of times the value has to be multiplied by 10. It is simply called base 10 logarithms.
Logarithms to the base e are called natural logarithms.

Previous Year Questions

The differential coefficient of f (Sinx) w.r.t. x where f(x)=logx is….[KCET 2004]
The differential coefficient of log10x with respect to logx10 is….….[KCET 2016]
If log72=?, then the value of log49(28) is….
The derivative of y = xsinx is...[UPSEE 2016]
The derivative of (logx)x with respect to log x is….
The sum of the divisors of 2⁴⋅3³⋅5³ is…..
The value of =dxd[xnlogaxex]=…
The value of dxd[xnlogaxex] is...[JKCET 2013]
If a=log23,b=log25,c=log72, then log14063 in terms of a, b, ca,b,c is…...[BITSAT 2007]
The general value of the real angle θ, which satisfies the equation,...[WBJEE 2019]

Sample Questions

Ques. Solve log ₃₂ (2 marks)

Ans. Since 32 can be expressed in terms of 2⁵ = 2*2*2*2*2

2⁵ = 32

Hence 5 is the exponent value

So log₃₂ = 5

Ques. Solve log₃(x+1) = 3 (2 marks)

Ans. We can rewrite the above one as (x+1) = 33

(x+1) = 27

X = 26

Therefore the solution for log₃(x+1) = 3 is 26

Ques. log(x+3) + log(x-1) = 1 (2 marks)

Ans. log[(x+3)(x+1)] = 1

log(x²+4x+4) = 1

(x²+4x+4) = 10¹

(x²+4x+4) = 0

(x+3)(x+1) = 0

Therefore x =-3, -1

Ques.Solve 6 ^2x = 4 (2 marks)

Ans. It can be written in the form of log 6 ^2x= log 4

2x log 6 = log 4

X = log 4/2 log 6

= 0.6020/2*0.7781

= 1.0823

Therefore the solution is 1.0823

Ques. Prove that log₂(x+2)+ log₂(4) = log (16) for x = 2 (2 marks)

Ans. Substitute 2 in the above equation we get

LHS = log₂(x+2)+ log₂(4)

= log₂(2+2)+ log₂(4)

= log₂(4)+ log₂(4)

= log₂(16)

= RHS

Therefore LHS= RHS

Hence proved

Ques. Find the x value from the given equation log₂(2x) = log₂(4x+7) (2 marks)

Ans. Given log₂(6x) = log₂(4x+2)

6x= 4x+2

2x = 2

X =1

The value of x in the equation log₂(2x) = log₂(4x+7) is 1.

Ques. Solve log(4x-3)-log(x-4) = log 5 (2 marks)

Ans. The above equation can be written in the form of

log(4x-3/x-4) = log 5

4x-3/x-4 = 5

4x-3 = 5x-20

X = 17

The x value of the above equation is 17

Ques. Solve 4^2x+1 = 21 and find the value of x (2 marks)

Ans. For the above equation apply log on both sides

Log 4^2x+1 = log 21

(2x+1) log 4 = log 21

2x+1 = (log 21)/(log 4)

= (1.3222)/(0.6020)

2x+1 = 2.19634

2x = 1.1963

X = 0.598

The value of x by solving the above equation is 0.598

Ques. Solve log₅(x-10) = 1 (2 marks)

Ans. The above equation can be written in the form of 5^-1 = x-10

5^-1 = x-10

5 = x-10

The value of x by solving the above equation is 15

Ques. Express 3logx+8log y = log b in Logarithmic free form. (2 marks)

Ans. Given 3logx+8log y = log b

Log x³ + log y⁸ = log b

log(x³ y⁸) = log b

x³ y⁸ = b

The above equation in logarithm free form is x³ y⁸ = b

Ques. Express log₁₀(3)+1 in the form of log₁₀ x (2 marks)

Ans. The above expression is written in the form of

= log₁₀(3)+1

= log₁₀(3)+ log₁₀(10)

= log₁₀(3*10)

= log₁₀(30)

The above expression in the form of log₁₀ x is log₁₀(30)

Ques. Express \((\frac{1}{3})^4 = \frac{1}{8}\) in logarithmic form. (2 marks)

Ans. By taking log of base \(\frac{1}{3}\) on both sides, we get

\((\frac {1}{3})^4 = \frac {1}{81}\)

⇒ \(log_{\frac{1}{3}}(\frac {1}{3})^4 =log_{\frac{1}{3}} \frac {1}{81}\)

⇒ \(4log_{\frac{1}{3}}(\frac {1}{3}) =log_{\frac{1}{3}} \frac {1}{81}\) (Since log_baⁿ = n log_ba)

⇒ 4 = \(log_{\frac{1}{3}} \frac {1}{81}\) (Since log_bb = 1)

⇒ \(log_{\frac{1}{3}} \frac {1}{81}\) = 4

Ques. Find the value of x satisfying log₁₀ (2x + x – 41) = x (1 – log₁₀?). (2 marks)

Ans. We have, log₁₀ (2x + x – 41) = x (1 – log₁₀?)

→ log₁₀(2x + x – 41)

→ x log₁₀2= log10 (2x )

→ 2 x + x – 41 = 2x

→ x = 41.

Check-More:

Class 11 & 12 PCMB Study Guides
NCERT Solutions for Class 11 Biology	NCERT Class 11 Biology Book	NCERT Solutions Class 6 to 12
NCERT Solutions for Class 12 Biology	Chemistry Study Notes	Important Chemistry Formulas
Topics for Comparison in Physics	Choice based questions in physics	Topics with relation in physics
Physics Study Notes	Class 12 Physics Book PDF	Class 12 Physics Practicals
Important Derivations in Physics	SI units in Physics	Important Physics Constants and Units
Physics MCQs	Formulas in Physics	Comparison topics in Chemistry
Comparison Topics in Biology	NCERT Solutions for Class 12 Maths	Class 12 Physics Notes
Biology Study Notes	Class 12 Biology Notes	Class 12 Maths Notes
Class 12 PCMB Notes	NCERT Class 12 Biology Book	NCERT Class 12 Maths Book
Chemistry MCQs	NCERT Solutions for Class 11 Maths	NCERT Solutions for Class 11 Physics
NCERT Solutions for Class 11 Chemistry	NCERT Solutions for Class 12 Chemistry	Class 12 Chemistry Practicals
Maths MCQs	NCERT Class 11 Physics Book	Class 12 Chemistry Notes
Biology MCQs	Important Maths Formulas	NCERT Solutions for Class 12 Physics

Logarithm Formula: Types, Rules, Solved Examples

What is a Logarithm?

Types of Logarithm

Common Logarithm

Natural Logarithm

Logarithm Rules

Logarithm Product Rule

Logarithm Division Rule

Logarithm Power Rule

Logarithm Zero Rule

Logarithm Identity Rule

Logarithm Exponential Rule

Change of Base Rule

Logarithm Formulas

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

2.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

Logarithm Formula: Types, Rules, Solved Examples

What is a Logarithm?

Types of Logarithm

Common Logarithm

Natural Logarithm

Logarithm Rules

Logarithm Product Rule

Logarithm Division Rule

Logarithm Power Rule

Logarithm Zero Rule

Logarithm Identity Rule

Logarithm Exponential Rule

Change of Base Rule

Logarithm Formulas

Things to Remember

Previous Year Questions

Sample Questions

CBSE CLASS XII Related Questions

1.Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

2. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

4. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6. Find: The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is:

Similar Mathematics Concepts

Comments

SUBSCRIBE TO OUR NEWS LETTER

1.
Using integration, find the area of the region bounded by the curve \( y = x|x| \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 2 \).

2.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

4.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6.
Find:

The principal value of \[ \sec^{-1}(\sqrt{2})+2\csc^{-1}(-2) \] is: