Natural Logarithm (ln)

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

• Natural logarithm (ln) definition
• Natural logarithm (ln) rules & properties
  • Derivative of natural logarithm (ln)
  • Integral of natural logarithm (ln)
• Natural logarithm (ln) table
• Natural logarithm calculator

Definition of natural logarithm

When

e ^y = x

Then base e logarithm of x is

ln(x) = log_e(x) = y

Natural logarithm rules and properties

Operation	Rule	Example
Multiplication	ln(x ∙ y) = ln(x) + ln(y)	ln(3 ∙ 7) = ln(3) + ln(7)
Division	ln(x / y) = ln(x) - ln(y)	ln(3 / 7) = ln(3) - ln(7)
Exponentiation	ln(x y) = y ∙ ln(x)	ln(28) = 8 ∙ ln(2)
Zero	ln(0) is undefined
Derivative	f (x) = ln(x) ⇒ f ' (x) = 1 / x
Integral	∫ ln(x)dx = x ∙ (ln(x) - 1) + C

 

Derivative of natural logarithm (ln) function

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 (ln) function

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

Natural logarithm table

x	ln x
0	undefined
0+	- ∞
0.0001	-9.210340
0.001	-6.907755
0.01	-4.605170
0.1	-2.302585
1	0.000000
2	0.693147
e	1.000000
3	1.098612
4	1.386294
5	1.609438
6	1.791759
7	1.945910
8	2.079442
9	2.197225
10	2.302585
100	4.605170
1000	6.907755
10000	9.210340

 

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See also

• Natural logarithm calculator
• Logarithm calculator
• Logarithm (log)
• e constant