Natural Logarithm – ln(x)

Natural logarithm is the logarithm to the base e of a number.

Definition of natural logarithm

When

e y = x

Then base e logarithm of x is

ln(x) = loge(x) = y

 

The e constant or Euler’s number is:

e ≈ 2.71828183

Ln as inverse function of exponential function

The natural logarithm function ln(x) is the inverse function of the exponential function ex.

For x>0,

f (f -1(x)) = eln(x) = x

Or

f -1(f (x)) = ln(ex) = x

Natural logarithm rules and properties

 

Logarithm product rule

The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.

logb(x ∙ y) = logb(x) + logb(y)

For example:

log10(3 ∙ 7) = log10(3) + log10(7)

Logarithm quotient rule

The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.

logb(x / y) = logb(x) – logb(y)

For example:

log10(3 / 7) = log10(3) – log10(7)

Logarithm power rule

The logarithm of x raised to the power of y is y times the logarithm of x.

logb(x y) = y ∙ logb(x)

For example:

log10(28) = 8∙ log10(2)

Derivative of natural logarithm

The derivative of the natural logarithm function is the reciprocal function.

When

f (x) = ln(x)

The derivative of f(x) is:

f ‘ (x) = 1 / x

Integral of natural logarithm

The integral of the natural logarithm function is given by:

When

f (x) = ln(x)

The integral of f(x) is:

∫ f (x)dx = ∫ ln(x)dx =
x ∙ (ln(x) – 1) + C

Ln of 0

The natural logarithm of zero is undefined:

ln(0) is undefined

The limit near 0 of the natural logarithm of x, when x approaches zero, is minus infinity:

Ln of 1

The natural logarithm of one is zero:

ln(1) = 0

Ln of infinity

The limit of natural logarithm of infinity, when x approaches infinity is equal to infinity:

lim ln(x) = ∞, when x→∞

Complex logarithm

For complex number z:

z = reiθ = x + iy

The complex logarithm will be (n = …-2,-1,0,1,2,…):

Log z = ln(r) + i(θ+2nπ) = ln(√(x2+y2)) + i·arctan(y/x))

Graph of ln(x)

ln(x) is not defined for real non positive values of x:

Natural logarithms table

x
ln x

0
undefined

0+
– ∞

0.0001
-9.210340

0.001
-6.907755

0.01
-4.605170

0.1
-2.302585

1
0

2
0.693147

e ≈ 2.7183
1

3
1.098612

4
1.386294

5
1.609438

6
1.791759

7
1.945910

8
2.079442

9
2.197225

10
2.302585

20
2.995732

30
3.401197

40
3.688879

50
3.912023

60
4.094345

70
4.248495

80
4.382027

90
4.499810

100
4.605170

200
5.298317

300
5.703782

400
5.991465

500
6.214608

600
6.396930

700
6.551080

800
6.684612

900
6.802395

1000
6.907755

10000
9.210340

 

Rules of logarithm ►

 

See also