Logs and Exponentials

The exponential function

expxex
appears widely in science. You need to be familiar with it, and with the companion function, the natural logarithm 
lny

Try the following exercises:

  1. Sketch the functions

    $$e^{x} \hspace*{0.5cm} \mbox{{\rm and}} \hspace*{0.5cm} \ln{y}$$

  2. The logarithm is said to be the inverse of the exponential; why?
  3. Give one reason why the exponential function appears so often.
  4. Write down expressions for the derivatives:

    $$\frac{d}{dx} e^x \hspace*{0.5cm} \mbox{{\rm and}} \hspace*{0.5cm} \frac{d}{dy} \ln y$$

    Satisfy yourselves that these expressions are consistent with what you see in your plots of the two functions.

  5. Sketch the following functions on the same graph

    $$e^{-x^2} \hspace*{0.5cm} \mbox{{\rm }} \hspace*{0.5cm} e^{-(x-2)^2} \hspace*{0.5cm} \mbox{{\rm }} \hspace*{0.5cm} e^{-4x^2}$$

  6. The chances of finding a molecule of gas moving with speed v fall off exponentially at large v through an equation of the form

    $$\mbox{\rm probability of speed close to }\,v \hspace*{0.5cm} \sim e^{-Bv^2}$$

    What can you say about B?

  7. Simplify the following expressions where possible:
    1. ex×e2x
    2. ex+e2x
    3. ln2x+ln3x
    4. ln[A+B]

Solution

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Solution

  1. Sketches of the functions:
    Graphic - Expoential Graph
    Graphic - Log Graph
  2. The logarithm and the exponential functions are the inverse of one another because each undoes the action of the other:

    Thus, for any z

    $$\ln \left[ \exp {z} \right ] =z \hspace*{0.5cm} \mbox{{\rm and}} \hspace*{0.5cm} \exp \left[ \ln {z} \right ] =z$$

    You can see this from your two sketches. How does it show up there?

  3. One reason why the exponential function appears so often is that the solution of the differential equation

    $$\frac{dy}{dt} = ky$$

    is

    y(t)=expkt

    The differential equation describes a process in which the rate of change of some quantity is proportional to the current value of that quantity.

    There are many processes like this; can you give an example?

  4. The derivatives are

    $$\frac{d}{dx} e^x = e^x \hspace*{0.5cm} \mbox{{\rm and}} \hspace*{0.5cm} \frac{d}{dy} \ln y = \frac{1}{y}$$

    You should be able to see this behaviour in your plots of ex and lny, remembering that the derivative of a function measures its slope .

  5. The three given functions look like this:

    Graphic - Bell Graphs

    The blue curve represents e-x2.

    The green curve represents e-(x-2)2

    Note that it is like the blue curve, but shifted along (by how much?)

    The red curve represents e-4x2

    Note that it is like the blue curve, but narrower (by how much?)

  6. The argument of an exponential (that is the z in expz) is always a pure number: it is dimensionless.

    It follows that the combination Bv2 is dimensionless.

    So

    $$B \hspace*{0.5cm} \mbox{{\rm must have units of}} \hspace*{0.5cm} \frac {1}{\mbox{speed}^2}$$

  7. The ‘simplifications’:
    1. ex×e2x=e3x
    2. ex+e2x is not usefully simplifiable
      WRONG!
      Some may have been tempted to write
      ex+e2x=e3x
      WRONG!
    3. ln2x+ln3x=ln6x2=ln6+2lnx
      WRONG!
      Some may have been tempted to write
      ln2x+ln3x=ln5x
      WRONG!
    4. ln[A+B] is not usefully simplifiable
      WRONG!
      Some may have been tempted to write
      ln[A+B]=lnA+lnB
      WRONG!