ln(x) = log_{e}(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 e^{x}.
For x>0,
f (f ^{-1}(x)) = e^{ln(x)} = x
Or
f ^{-1}(f (x)) = ln(e^{x}) = 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(2^{8}) = 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) = ∞ ,when x→∞ | |
Euler's identity | ln(-1) = iπ |
Logarithm product rule
The logarithm of the multiplication of x and y is the sum of logarithm of x and logarithm of y.
log_{b}(x ∙ y) = log_{b}(x) + log_{b}(y)
For example:
log_{10}(3 ∙ 7) = log_{10}(3) + log_{10}(7)
Logarithm quotient rule
The logarithm of the division of x and y is the difference of logarithm of x and logarithm of y.
log_{b}(x / y) = log_{b}(x) - log_{b}(y)
For example:
log_{10}(3 / 7) = log_{10}(3) - log_{10}(7)
Logarithm power rule
The logarithm of x raised to the power of y is y times the logarithm of x.
log_{b}(x ^{y}) = y ∙ log_{b}(x)
For example:
log_{10}(2^{8}) = 8∙ log_{10}(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 = re^{iθ} = x + iy
The complex logarithm will be (n = ...-2,-1,0,1,2,...):
Log z = ln(r) + i(θ+2nπ) = ln(√(x^{2}+y^{2})) + 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 |
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