DERIVATIVE OF THE NATURAL LOGARITHM


y = ln(x)
Graph
The exponential function has an inverse function, which is called the natural logarithm, and is denoted ln(x). Our goal on this page is to verify that the derivative of the natural logarithm is a rational function. Specifically, we show that the following is true.

Dx(ln(x)) = 1/x


In order to conclude that the exponential function even has an inverse function, we recall that the graph of the exponential function passes the Horizontal Line Test: no horizontal line in the x-y plane touches the graph of y=ex more than once. (Symbolically, this means that if ex=ex', then x=x'.) The exponential function therefore has an inverse function, and the graph of this inverse function is obtained from the graph of the exponential function by reflection across y=x. (See "Horizontal Line Test" on the page "Inverse Functions.")


Basic properties of the natural logarithm are derived from properties of the exponential function and general facts about inverse functions.

Properties of the Natural Logarithm
  • The domain of the natural logarithm is the set of all positive real numbers.
    (You can't take the log of a negative number!)
  • The image of the natural logarithm is the set of all real numbers.
  • The natural logarithm is differentiable.
  • For any positive real number a and any real number x,
    • ln(a) = x if and only if ex = a;
    • eln(a) = a;
    • ln(ex) = x;
    • ln(ax) = xln(a).
  • For any positive real numbers a and b, ln(ab) = ln(a) + ln(b).
  • ln(1) = 0 and ln(e) = 1.

  • Since the exponential function is differentiable and is its own derivative, the fact that ex is never equal to zero implies that the natural logarithm function is differentiable. (See "Derivatives of Inverse Functions.")

    We can compute the derivative of the natural logarithm by using the general formula for the derivative of an inverse function. Simply set f(x)=ex, so that f '(x)=ex and f -1(x)=ln(x).

    Dx(ln(x)) = Dx(f -1(x)) = 1/f '(f -1(x)) = 1/eln(x) = 1/x

    Alternatively, we can use the chain rule.
    eln(x) = x

    Dx(eln(x)) = Dx(x)

    eln(x)[Dx(ln(x))] = 1

    Dx(ln(x)) = 1/eln(x) = 1/x



    We can now explore the derivatives of all of the exponential and logarithmic functions.


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    William A. Bogley
    Robby Robson