## Exponential Graphs

Exponential graphs are those of the form for positive a. You can sketch the graph of for positive a by considering the y coordinates that correspond to various x values. Graphs of this form will always cross the y-axis at 1 since for any a.

The following table shows coordinates for the graph for x taking integer values between -3 and 3:

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

y | 8 | 4 | 2 | 1 | 0.5 | 0.25 | 0.125 |

It follows that the graph of for will have a shape like .

The graph of for will have a shape like the graph above but will be reflected in the y-axis. This is because when you multiply a number less than 1 by itself, it becomes smaller.

The following table shows coordinates for the graph for x taking integer values between -3 and 3:

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
---|---|---|---|---|---|---|---|

y | 0.125 | 0.25 | 0.5 | 1 | 2 | 4 | 8 |

For any number , the graph will have the same shape as .

For a=1, the graph of is the horizontal line . This is because you are calculating 1 to any power, which is always 1. For negative a, fractional powers become an issue and complex numbers need to be considered.

The diagram shows the graph of where e, sometimes known as Euler’s number, is given by e=2.718281828459… Since e is positive and greater than 1, it looks very similar to the first graph above.

The number e is special because everywhere on this graph, the gradient is the same as the y-coordinate. See differentiating e to the x.

## Logarithmic vesus Exponential Graphs

As well as exponential graphs, there are logarithmic graphs. is considered to be the inverse of – see more on logs. It follows that (blue solid line) is the inverse of (red dotted line) and so their graphs are reflections of each other in the line y=x (green dotted line). Since and are mathematical inverses we have that .

Specifically, the **natural logarithm** is the logarithm that corresponds to e. That is, given an equation of the form , it can be said that . Since e is a special number, log to the base e has its own name. That is, it is the natural logarithm and often called ln so is more often written as .

and are mathematical inverses and we have that . Notice that for any positive a (including ln) cannot be evaluated for negative x – see more on logs.

**YouTube** – Graphing Logarithmic Functions

### Estimating Parameters for

Consider the equation . Note that, according to BIDMAS, this is x to the power of n, then multiplied by a. This is a stretch to a standard polynomial curve – see Curve Sketching. Given this relationship and a dataset that approximately fits it, it is possible to estimate the parameters a and n. First consider what happens when logging both sides:

Note that the bases are missing this is true for any base (provided the same base is used for both). In the same way that you can plot y against x, it is possible to plot log(y) against log(x). Recall that, in the equation y=mx+c, m is the gradient and c is the y-intercept. In addition, we can write log(y) as nlog(x)+log(a) and so, in the plot of log(y) against log(x), n is the gradient and log(a) is the y-intercept.

#### Example

The following table follows the relationship where the y values are given to one decimal place. By plotting against , this allows us to estimate the parameters a and n to 1 decimal place.

x | 2 | 3 | 4 | 5 |
---|---|---|---|---|

y | 26.4 | 104.7 | 278.6 | 594.9 |

### Estimating Parameters for

Now consider the equation . Like the above, this is an exponential curve, provided b is positive. Similarly to before, given a dataset or similar, we could estimate the parameters b and k. Taking logs:

It follows that log(y) can be written as log(b)x+log(k) and so, this time, in the plot of log(y) against x, log(b) is the gradient and log(k) is the y-intercept.

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