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Mathematics Dictionary
Dr. K. G. Shih
Logarithm ln(x)


  • Q01 | - Properties of y = ln(x)
  • Q02 | - Find derivative of y = ln(x)
  • Q03 | - Series of y = ln(1 + x)
  • Q04 | - Series of y = ln(1 - x)
  • Q05 | - Formula

    • Q01. Properties of y = ln(x)

    Diagram of y = ln(x)     Deifintion of y = ln(x)
    • If y = e^x then x = ln(y)
    • If y = ln(x) then x = e^(ln(y))
    • Composite function
      • ln(e^x) = x
      • e^(ln(x)) = x.
    • Value of y is between 0 and +infinite
    Properties of Y = e^x
      1. Domain : real values of x > 0
      2. Range : -infinite to +infinite
      3. The curve is always increasing (y' > 0) 4. The curve is concave downward (y" < 0) 5. It is the inverse of y = e^x
    Special values
    • ln(1) = 0
    • ln(e) = 1

    Go to Begin

    Q02. Find derivative of y = ln(x)

    Find y'
    • y = ln(x), 1st serivative = 1/x
    • y = ln(x), 2nd serivative = -1/(x^2)
    • y = ln(x), 3rd serivative = +2/(x^3)

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    Q03. Series of ln(1 + x)
    Binomial Theory
    • (1 - x)^n = 1 + C(n,1)*(-x) + C(n,2)*(-x)^2 + ....
    • Let n = -1, then
      • C(n, 1) = n = -1
      • C(n, 2) = n*(n - 1)/(2!) = 1
      • C(n, 3) = n*(n - 1)*(n - 2) = -1
    • 1/(1 - x) = 1 - (-x) + ((-1)*(-1 - 1)/(2!))*((-x)^2) + ....
    • = 1 + x + x^2 + x^3 + .....
    Series of ln(1 - x)
    • Since ∫[1/(1 - x)]dx = -ln(1 - x)
    • Using binomial theory we have
    • arctan(x) = ∫[1/(1 - x)]dx
    • = ∫[1 + x + x^2 + x^3 + ....]dx
    • = x + (x^2)/2 + (x^3)/3 + .....
    Go to Begin

    Q04. Series of ln(1 + x)

    Binomial Theory
    • (1 + x)^n = 1 + C(n,1)*(x) + C(n,2)*(x)^2 + ....
    • Let n = -1, then
      • C(n, 1) = n = -1
      • C(n, 2) = n*(n - 1)/(2!) = 1
      • C(n, 3) = n*(n - 1)*(n - 2) = -1
    • 1/(1 - x) = 1 - (x) + ((-1)*(-1 - 1)/(2!))*((x)^2) + ....
    • = 1 - x + x^2 - x^3 + .....
    Series of ln(1 + x)
    • Since ∫[1/(1 + x)]dx = ln(1 + x)
    • Using binomial theory we have
    • arctan(x) = ∫[1/(1 + x)]dx
    • = ∫[1 - x + x^2 - x^3 + ....]dx
    • = x - (x^2)/2 + (x^3)/3 - .....

    Go to Begin

    Q05. Formula
    • e^(ln(x)) = x
    • ln(e^x) = x
    • If y = ln(x), y' = 1/x
    • If y = ln(x), y" = -1/(x^2)
    • ∫[1/x]dx = ln(x)

    Go to Begin
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