2. Graphs of Exponential `y=b^x`, and Logarithmic `y=log_b x` Functions

by M. Bourne

Exponential Growth

We saw an example of an exponential growth graph (showing how invested money grows over time) at the beginning of the chapter.

The exponential curve is especially important in mathematics. Exponential growth and decay are common events in science and engineering and it is valuable if you know and recognise the shape of these curves.

Example 1

Sketch the graph of `y = 10^x`.

Answer

Exponential Decay

Radioactive decay is the most common example of exponential decay. Here we have 100 g of radioactive material decaying over time.

Notice that the function value (the y-values) get smaller and smaller as x gets larger (but the curve never cuts through the x-axis.). Also notice that the slope of the curve is always negative, but gets closer to 0 as x increases.

Since the amount of radioactive material becomes less over time, and the amount we are talking about becomes meaningless, we normally talk about the half life, that is, the amount of time it takes for the substance to reduce to half of its original mass. In our example, it takes about 6.5 minutes for the 100 g of stuff to decay to 50 g.

You can see another application of exponential decay in the differential equations section Application: Series RC Circuit. As the capacitor becomes fully charged, the current drops to zero. (Don't be scared by the complicated-looking mathematics in that section...)

Graph of the Logarithmic Function

Example 2

Sketch the graph of `y = log_10 x`.

Answer

Exponential Functions and Logarithmic Functions are Inverses

NOTE: The two functions `f(x) = 10^x` and `f(x) = log\ x` are on the same button on your calculator because they are inverses of each other (like e^x and `ln\ x` also.)

If we plot them on the same axes, we see that they are just reflections of each other in the line y = x.

In this graph, `f(x) = 10^x` is in green and `f(x) = log\ x` is in magenta, and we can see they are reflections of each other in y = x (dashed, grey).

Exercises

1. The velocity of a certain falling object (which is being affected by air resistance) is given by:

v = 95(1 − e^-0.1t) where v is in km/h and t is the time of fall in seconds.

Sketch the graph.

Answer

We come across the same kind of graph again later, in the section on electronics in differential equations, Application: Series RL Circuit, where the current builds up in an inductor.

2. Plot the graph of: `y = 3\ log_2 x`

Answer