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Natural Logarithm (ln)

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

Natural logarithm rules and properties

OperationRuleExample
Multiplication

ln(x ∙ y) = ln(x) + ln(y)

ln(37) = 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) = 8ln(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

xln x
0undefined
0+- ∞
0.0001-9.210340
0.001-6.907755
0.01-4.605170
0.1-2.302585
10.000000
20.693147
e1.000000
31.098612
41.386294
51.609438
61.791759
71.945910
82.079442
92.197225
102.302585
1004.605170
10006.907755
100009.210340

 


See also

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