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Natural Log in mathematics is a way of representing the exponents. We know that a logarithm is always defined with abase and for the natural log, the base is “e”. The natural log is used for solving various problems of Integration, Differentiation, and others.

The natural log formula is given as, suppose, ex = a then loge = a, and vice versa. Here loge is also called a natural log i.e., log with base e. The natural log is always represented by the symbol “ln”. Thus, ln x = loge x. In this article, we will learn about Natural log, Natural Log Formula, Examples, and others in detail.

What is Natural Log?

Natural log is the log of a number with base “e” where ‘e’ is Euler number and its value is 2.718 (approximately). The natural log is defined by the symbol ‘ln’.

For example, the natural log of a positive number is ‘ln x’. The natural log of numbers is a way of representing an exponent. Suppose we are given the exponent ex then its natural log is ln x.

Natural Log Definition

Natural log of any number is defined as the way of representing an exponent. Let take an exponent,

ex = y

Then natural log of number is,

y = ln (x)

The image added below explains the definition of the log.

Log-Definition

Natural Log Formula

Natural log of a number is the other of representing a number. Various natural log formulas are,

  • ln (1) = 0
  • ln (e) = 1
  • ln (-x) = Not Defined {log of negative number is Not-Defined}
  • ln (∞) = ∞
  • ln(ex) = x, x ∈ R

Product Rule

When we have a natural log of the product of two numbers, then it can be represented as the addition of the natural log of the first number and the natural log of the second number.

ln(xy) = ln x + ln y

Quotient Rule

When we have a natural log of a fraction of two numbers, then it can be represented as the subtraction of natural log of the first number and the natural log of the second number.

ln(x/y) = ln x – ln y

Power Rule

When we have a natural log of x to power r, then it can be represented as r times ln x

ln(xr) = r.ln x

Reciprocal Rule

When we have a natural log of reciprocal of x, it can be represented as minus of the natural log of x.

ln(1/x) = -ln x

Change of Base

Base of log can be easily changed using the formula,

loge a = (logc a)/(logc e)

Natural Log Formulae Table

Representation of Natural Logloge x = ln x 
ln (1)ln 1 = 0
ln (e)ln e = 1
ln (-x)Not defined
ln (∞)∞ 
Conversion Formula

ln x = y ⇔ ey = x

eln xx ,  x>0
ln (ex)x , x ∈ R
Product Ruleln(xy) = ln x + ln y
Quotient Ruleln(x/y) = ln x – ln y
Power Ruleln(xr) = r.ln x
Reciprocal Ruleln(1/x) = -ln x
Base change Rulelogba = (ln a)/(ln b)
Equality of lnln x = ln y ⇔ x = y

Read more about Log Rules.

Natural Logarithms Table

The natural log of any number is the log with base e. The natural log of various number are added in the table below,

x

ln (x)

0

Undefined

0.1

-2.302585

1.0

0.000000

2.0

0.693147

e (≈ 2.7183)

1.000000

3.0

1.098612

4.0

1.386294

5.0

1.609438

6.0

1.791759

7.0

1.945910

8.0

2.079442

9.0

2.197225

10.0

2.302585

20.0

2.995732

30.0

3.401197

50.0

3.912023

100.0

4.605170

Difference Between Log and Ln

Difference between Log and Ln is added in the table given below,

log

ln

Base of log is 10.

Base of ln is ‘e’.

It is represented as log (x)

It is represented as ln (x)

For logarithim,

10x = y

For ln,

ex = y

Example: log10 (10) = 1

Example: ln (10) = loge (10) = 2.3025

Natural Log Derivatrive

Natural log derivative is the derivative of ln x. The derivative of ln x is x. i.e.

d/dx {ln (x)} = 1/x

Natural Log Integration

Natural log integration is the integration of ∫ ln(x) dx. Integration of ln (x) can not be easily calculated using normal integration formula, it is calculated by taking the ILATE rule as, such that ∫ 1.ln(x) dx, now integration of this can be calculated, the integration of ln (x) is,

∫ ln(x) dx = x·ln(x) – x + C

Natural Lag Laws

Various rules associated to the natural log are,

Product Rule

  • loge (XY) = loge (X) + loge (Y)

Quotient Rule

  • loge (X/Y) = loge (X) – loge (Y)

Zero Rule

  • loge (1) = 0

Identity Rule

  • loge (e) = 1

Read More,

Examples Using Natural Log Formula

Example 1: Solve,

  • ex = 10  
  • ln x = 2  
  • eln 15  
  • ln(e29)  
  • ln(39)  
  • ln(15/4) 
  • ln(39
  • log57

Solution:

1: ex = 10

x = ln 10

x = 2.303

2: ln x = 2

x = e2

x = 7.389

3: eln 15 = 15 

4: ln(e29) = 29

5: ln (39)

= ln(13 × 3)

= ln 13 + ln 3

= 2.565 + 1.099 = 3.664

6: ln (15/4)

= ln 15 – ln 4

= 2.708 – 1.386 = 1.322

7: ln(39)

= 9 × ln 3

= 9 × 1.099 = 9.891

8: log5 7

= (ln 7)/(ln 5)

= 1.946/1.609 = 1.209

Example 2: Solve, ln(15x – 3) = 2

Solution: 

ln(15x – 3) = 2

15x – 3 = e2

15x -3 = 7.389

15x = 10.389

x = 10.389/15 ⇒ x = 0.6926

Example 3: If 8exy + 2 = 98  and  2ez + 3 = 79, then find the value of x + y,  where z = x2 + y2

Solution: 

8exy + 2 = 98

8exy = 98-2 = 96

exy = 96/8 = 12

ln(exy) = ln 12

xy = 2.4849…(i)

2ez + 3 = 79

2ez = 79-3 = 76

ez = 76/2 = 38

ln(ez) = ln 38

z = 3.6375…(ii)

z = x2 + y2

Now, (x + y)2 = x2 + y2 + 2xy

From eq(i) and eq(ii)

(x + y)2 = 3.6375 + 2 × 2.4849

(x + y)2 = 3.6375 + 4.9698

(x + y)2 = 8.6073

(x + y) = √8.6073

x + y = 2.933

Example 4: Simplify y = ln 25 – ln 15

Solution: 

y = ln 25 – ln 15

y = ln(5 × 5) – ln(5 × 3) 

y = ln 5 + ln 5 – [ln 5 + ln3]  

y = ln 5 + ln 5 – ln 5 – ln3  

y = ln 5 – ln 3

y = ln (5/3)

y = 0.511 

Example 5: Solve: ln(e15) + e 2+x = 16

Solution: 

ln(e15) + e 2+x = 16

⇒ 15 + e2+x = 16

⇒ e2+x = 16 – 15

⇒ e2+x = 1

⇒ ln( e2+x )= ln 1

⇒ 2 + x = 0

x = -2

Example 6: Evaluate: p = log35 – log36 + log310

Solution:

Using base change of log formula,

p = (ln 5/ ln 3) – (ln 6/ ln 3) + (ln 10/ ln 3)

p = [ln 5 -(ln 6 + ln 10)] / ln 3 

p = [ln 5 – ln (6 × 10)]/ ln 3 

p = [ln 5 – ln 60]/ ln 3 

p = [ln(5/60)] / ln 3

p = [ln(1/12)] / ln 3

p = [ln (12)-1] / ln 3

p = [-1×ln 12] / ln 3

p = -ln 12 / ln 3

p = -2.262

FAQs on Natural Log

1. What is Natural Log in Maths?

Natural Log in mathematics is the way of representing exponents. It is log of a number with base ‘e’. It is represented by symbol ‘ln’. Suppose we are given an exponent,

y = ex

Then in exponent form it is represented as,

ln (y) = x

2. What is Natural Log of 2?

Natutal log of 2 or ln (2) is equal to 0.69314, i.e.

ln (2) = 0.69314

3. What is Natural Log of 1?

Natutal log of 1 or ln (1) is equal to 0, i.e.

ln (1) = 0

4. How is Natural Log of x Represented?

Natural log of x is represented as ln (x)

5. What is Natural log of Infinity?

Natutal log of ∞ or ln (∞) is equal to 1, i.e.

ln (∞) = ∞

6. What is Natural Log Derivative?

Natural log derivative is represented d/dx {ln (x)} as,

d/dx {ln (x)} = 1/x

7. What is Natural Log Integration?

Natural log integration is represented ∫{ln (x)} as,

∫ {ln (x)} dx = x·ln(x) – x + C

8. What is Natural Log of e?

Natutal log of e or ln (e) is equal to 1, i.e.

ln (e) = 1

9. What is Natural Log base?

The base of ntural log is ‘e’ or Euler Numbers. ‘e’ is a irrational number and its value is ‘e = 2.718’



Last Updated : 10 Jan, 2024
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