The Exponential Function – Why so useful?

An exponential function is a function where the input variable (usually written as x) is an exponent. Functions with constants in the exponent are also considered exponential functions.

exponential_function_two_to_x

An exponential function

Then what is so amazing about these exponential functions? It all comes down to the rate of change of such a function (also known as the derivative of the function). Exponential functions are the only functions where the derivative of the function is directly proportional to the value of the function at any given point. This makes these functions extremely useful in mathematical modelling of real life situations, as in the real world, for example, growth of population is higher for higher populations.
To prove that an exponential function is proportional to it’s derivative, you must first differentiate the function. Re-writing the function as I’ve done in step 1 allows for a simple way of differentiating the function.

derivative1

Above, in step 2, I have just applied the chain rule, given that we have a compound function.

derivative2

In the steps which follow (shown above) I’ve used the fact that the derivative of e^x =  e^x, and the product rule, to finish the derivative. The final step was just to simplify the final answer by undoing the first step.

So what has this shown? The derivative of an exponential function is directly proportional to its value at any point, and the constant of proportionality is lnb (the natural logarithm of the base).

From the proof above we can say if lnb = 1 the derivative of our exponential function will be equal to our function!! Is there therefore a value for b where the constant lnb =1? Yes, there is.

lnb

Here “e” refers to the number 2.718… often called Euler’s number or simply the base for natural logarithms. If you replace b in our general formula for “e”, you get an exponential function where d/dx (f(x)) = f(x). This function, with exponent base “e” is called THE exponential function.

theefunc

This function (shown above) has so many physical applications, that it is impossible to write about these in just one article. The list of things in nature who’s growth or decay are directly proportional to the function which describes them is endless! However I will attempt to mention a few without subjecting you to thousands upon thousands of words.

In the real world, it must be noted, most things are modeled by negative exponential functions. This is because e^x rises to infinity, and in the finite world we live in this isn’t appropriate. The function e^-x tends towards 0 without ever reaching it. As you will discover this is the reality for many real systems. the function (1-e^-x) describes growth really well, as it has it growing very quickly at first, but then tending towards a fixed value.

Capacitors

In electronics capacitors are components which store charge. They consist of two parallel plates, made of electric conductors, separated by a dialectric. A capacitor in a circuit has a positive and negatively charged plate, with the positively charged plate usually pointed to the negative terminal of the power supply.

Various capacitors

When a potential difference is applied electrons flow to the positive plate, building up there as they are repelled by the negative plate. As they do this the current in the circuit falls. The potential difference across the capacitor rises, as does the charge stored in the capacitor. When the capacitor is discharged through a fixed resistor, the opposite happens.

Both the rise and fall of all these variables (current, charge, p.d) can be modeled by the exponential function. For example when charging a capacitor, the voltage across the capacitor increases, as a negative exponential function:

voltagemate

When a capacitor is discharging the following occurs:

dischargemate

The equations above attached to the graphs show that the Voltage at any time is a certain percentage of the total or final voltage. That percentage growth or decay is defined by the exponential function, with the variable being time- and a constant involved (-1/RC). The exponential function models perfectly what happens with capacitors used everyday.

Other physical applications of the exponential function:

  • Nuclear Decay
  • Damped Oscillations
  • Photon absorption
  • First order reaction rates
  • Boltzmann distribution
  • Terminal Velocity
  • + many more

Other applications of the exponential function:

  • Population growth
  • Compound interest
  • Mortgages
  • + many more

In conclusion there are endless applications of the exponential function, which so brilliantly models many things we interact with on a day to day basis. It’s elegance in doing so is so brilliant. Without the exponential function everything would be so much more difficult, with complex polynomials and exponentials attempting to recreate its modelling superiority.

Thank you the exponential function- you’re e^xcellent.

-PlanckTime Amin

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