Online Reference

 

 

ALGEBRA
RAPID TABLES

 

 

    Home > Math > Algebra > Natural logarithm

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:

e2.71828183

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

Rule nameRuleExample
Product rule

ln(x ∙ y) = ln(x) + ln(y)

ln(37) = ln(3) + ln(7)

Quotient rule

ln(x / y) = ln(x) - ln(y)

ln(3 / 7) = ln(3) - ln(7)

Power rule

ln(x y) = y ∙ ln(x)

ln(28) = 8ln(2)

Ln of zero

ln(0) is undefined

 

 
Ln of one

ln(1) = 0

 
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 logarithms table

xln x
0undefined
0+- ∞
0.0001-9.210340
0.0010-6.907755
0.0100-4.605170
0.1000-2.302585
1.00000.000000
2.00000.693147
e2.71831.000000
3.00001.098612
4.00001.386294
5.00001.609438
6.00001.791759
7.00001.945910
8.00002.079442
9.00002.197225
10.00002.302585
20.00002.995732
30.00003.401197
40.00003.688879
50.00003.912023
60.00004.094345
70.00004.248495
80.00004.382027
90.00004.499810
100.00004.605170
200.00005.298317
300.00005.703782
400.00005.991465
500.00006.214608
600.00006.396930
700.00006.551080
800.00006.684612
900.00006.802395
1000.00006.907755
10000.00009.210340

 

Natural logarithm calculator

 


See also

© 2006-2008 RapidTables.com