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
| Rule name | Rule | Example |
|---|
Product rule
|
ln(x ∙ y) = ln(x) + ln(y)
|
ln(3 ∙ 7) = 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) = 8∙ ln(2)
|
ln derivative
|
f (x) = ln(x) ⇒ f ' (x) = 1 / x |
|
ln integral
|
∫ ln(x)dx = x ∙ (ln(x) - 1) + C |
|
ln of negative number
|
ln(x) is undefined when x≤ 0 |
|
ln of zero
|
ln(0) is undefined |
|
| |
|
ln of one
|
ln(1) = 0 |
|
ln of infinity
|
lim ln(x) = ∞ ,whenx→∞ |
|
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→∞
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 |
| 0.0010 |
-6.907755 |
| 0.0100 |
-4.605170 |
| 0.1000 |
-2.302585 |
| 1.0000 |
0.000000 |
| 2.0000 |
0.693147 |
| e ≈ 2.7183 |
1.000000 |
| 3.0000 |
1.098612 |
| 4.0000 |
1.386294 |
| 5.0000 |
1.609438 |
| 6.0000 |
1.791759 |
| 7.0000 |
1.945910 |
| 8.0000 |
2.079442 |
| 9.0000 |
2.197225 |
| 10.0000 |
2.302585 |
| 20.0000 |
2.995732 |
| 30.0000 |
3.401197 |
| 40.0000 |
3.688879 |
| 50.0000 |
3.912023 |
| 60.0000 |
4.094345 |
| 70.0000 |
4.248495 |
| 80.0000 |
4.382027 |
| 90.0000 |
4.499810 |
| 100.0000 |
4.605170 |
| 200.0000 |
5.298317 |
| 300.0000 |
5.703782 |
| 400.0000 |
5.991465 |
| 500.0000 |
6.214608 |
| 600.0000 |
6.396930 |
| 700.0000 |
6.551080 |
| 800.0000 |
6.684612 |
| 900.0000 |
6.802395 |
| 1000.0000 |
6.907755 |
| 10000.0000 |
9.210340
|
:: موضوعات مرتبط:
فرمول های مفید ریاضی ,
,
:: برچسبها:
ln ,
لگاریتم ,
فرمول ,
ریاضی ,
نپر ,
طبیعی ,
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