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.
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)
Representation of Natural Log |
loge 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 x |
x , Â x>0 |
ln (ex) |
x , x ∈ R |
Product Rule |
ln(xy) = ln x + ln y |
Quotient Rule |
ln(x/y) = ln x – ln y |
Power Rule |
ln(xr) = r.ln x |
Reciprocal Rule |
ln(1/x) = -ln x |
Base change Rule |
logba = (ln a)/(ln b) |
Equality of ln |
ln 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,
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,
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
Identity Rule
Read More,
Example 1: Solve,
- ex = 10 Â
- ln x = 2 Â
- eln 15Â Â
- ln(e29) Â
- ln(39) Â
- ln(15/4)Â
- ln(39)Â
- log57
Solution:
1: ex = 10
x = ln 10
x = 2.303
2: ln x = 2
x = e2
x = 7.389
3: eln 15 = 15Â
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:Â
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:Â
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:Â
y = ln 25 – ln 15
y = ln(5 × 5) – ln(5 × 3)Â
y = ln 5 + ln 5 – [ln 5 + ln3] Â
y = ln 5 + ln 5 – ln 5 – ln3 Â
y = ln 5 – ln 3
y = ln (5/3)
y = 0.511Â
Example 5: Solve: ln(e15) + e 2+x = 16
Solution:Â
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 3Â
p = [ln 5 – ln (6 × 10)]/ ln 3Â
p = [ln 5 – ln 60]/ ln 3Â
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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