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

by M. Bourne

## Exponential Growth

### Need Graph Paper?

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

Substituting into a table of values gives us:

x |
`-1` | `-0.5` | `0` | `0.5` | `1` | `1.5` | `2` |

y |
`0.1` | `0.316` | `1` | `3.16` | `10` | `31.6` | `100` |

We plot these points to give:

Graph of `y=10^x`.

**Note** that the curve passes through `(0, 1)` (on the *y*-axis). For negative `x`-values, the graph gets very close to the `x`-axis, but doesn't touch it.

This is an **exponential growth curve**, where the *y*-value increases and the slope of the curve increases as *x* increases.

## 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

We could take some typical values and join the dots to graph the log function, as follows:

x |
`0.01` | `0.1` | `0.5` | `1` | `2` | `3` | `4` |
---|---|---|---|---|---|---|---|

f(x) |
`-2` | `-1` | `-0.30` | `0` | `0.30` | `0.48` | `0.60` |

Graph of `y=log_10 x`.

Notice that we cannot take **zero** or **negative** values for *x*. Can you figure out why not?

Our curve passes through `(1, 0)` (on the *x*-axis).

## 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).

Graphs of `y=10^x` (green) and `y=log_10 x` (magenta).

### 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}) wherevis in km/h andtis the time of fall in seconds.

Sketch the graph.

Answer

A table of values gives us:

t |
`0` | `10` | `20` | `30` | `40` | `50` |

v |
`0` | `60.1` | `82.1` | `90.3` | `93.3` | `94.4` |

The sketch is:

Graph of *v* = 95(1 − *e*^{-0.1t}).

If we take larger values of *t*, the velocity does not go over `95`. In this case, the **terminal velocity** of the object is `95` km/h (marked on the graph as a dotted magenta line).

While this looks a bit like the graph of the logarithm function, it is quite different. This one starts at `(0, 0)`, does not pass through `(1, 0)` and does not increase without bound.

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

We do not have a `log_2 x` button on our calculators. The only way we can do this for now is to draw up a table of values and plot the points, or use a computer.

Later we will see how to plot this using
the** change of base formula** which we meet in 5. Logs to the Base *e*.

Using a computer to graph it gives us:

Graph of `y=3log_2 x`.

Unlike the graph in the previous example, this graph does not have a limiting value as *x* increases.