W4.1 Logs and Exponentials
The exponential function
Try the following exercises:
Sketch the functions
- The logarithm is said to be the inverse of the exponential; why?
- Give one reason why the exponential function appears so often.
Write down expressions for the derivatives:
Satisfy yourselves that these expressions are consistent with what you see in your plots of the two functions.
Sketch the following functions on the same graph
The chances of finding a molecule of gas moving with speed v fall off exponentially at large v through an equation of the form
What can you say about B?
- Simplify the following expressions where possible:
- ex×e2x
- ex+e2x
- ln2x+ln3x
- ln[A+B]
Solution
Reveal

Solution
- Sketches of the functions:
The logarithm and the exponential functions are the inverse of one another because each undoes the action of the other:
Thus, for any z
You can see this from your two sketches. How does it show up there?
One reason why the exponential function appears so often is that the solution of the differential equation
is
y(t)=expktThe 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?
The derivatives are
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 .
The three given functions look like this:
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?)
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
- The ‘simplifications’:
- ex×e2x=e3x
- ex+e2x is not usefully simplifiable
WRONG! Some may have been tempted to write ex+e2x=e3xWRONG! - ln2x+ln3x=ln6x2=ln6+2lnx
WRONG! Some may have been tempted to write ln2x+ln3x=ln5xWRONG! - ln[A+B] is not usefully simplifiable
WRONG! Some may have been tempted to write ln[A+B]=lnA+lnBWRONG!