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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
- 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 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
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
Natural logarithm calculator ►
See also
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