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.
f (f -1(x)) = eln(x) = x
f -1(f (x)) = ln(ex) = x
ln(x ∙ y) = ln(x) + ln(y)
ln(3 ∙ 7) = ln(3) + ln(7)
ln(x / y) = ln(x) - ln(y)
ln(3 / 7) = ln(3) - ln(7)
ln(x y) = y ∙ ln(x)
ln(28) = 8∙ ln(2)
|f (x) = ln(x) ⇒ f ' (x) = 1 / x|
|∫ ln(x)dx = x ∙ (ln(x) - 1) + C|
ln of negative number
|ln(x) is undefined when x ≤ 0|
|ln(0) is undefined|
|ln(1) = 0|
|lim ln(x) = ∞ ,when x→∞|
|Euler's identity||ln(-1) = iπ|
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)
log10(3 ∙ 7) = log10(3) + log10(7)
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)
log10(3 / 7) = log10(3) - log10(7)
The logarithm of x raised to the power of y is y times the logarithm of x.
logb(x y) = y ∙ logb(x)
log10(28) = 8∙ log10(2)
The derivative of the natural logarithm function is the reciprocal function.
f (x) = ln(x)
The derivative of f(x) is:
f ' (x) = 1 / x
The integral of the natural logarithm function is given by:
f (x) = ln(x)
The integral of f(x) is:
∫ f (x)dx = ∫ ln(x)dx = x ∙ (ln(x) - 1) + C
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:
The natural logarithm of one is zero:
ln(1) = 0
The limit of natural logarithm of infinity, when x approaches infinity is equal to infinity:
lim ln(x) = ∞, when x→∞
For complex number z:
z = reiθ = x + iy
The complex logarithm will be (n = ...-2,-1,0,1,2,...):
Log z = ln(r) + i(θ+2nπ) = ln(√(x2+y2)) + i·arctan(y/x))
ln(x) is not defined for real non positive values of x:
|e ≈ 2.7183||1|
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