Natural Logarithm - ln(x)
Natural logarithm is the logarithm to the base e of a number.
- Natural logarithm (ln) definition
- Natural logarithm (ln) rules & properties
- Graph of ln(x)
- Natural logarithms (ln) table
- Natural logarithm calculator
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) = ∞ , when x→∞
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
See also
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