# Exponential Graphs

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

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

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 -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 taking integer values between -3 and 3:

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

For , 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 , sometimes known as Euler’s number, is given by … Since is positive and greater than 1, it looks very similar to the first graph above.

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

## Logarithmic 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 (green dotted line). Since and are mathematical inverses we have that .
Specifically, the natural logarithm is the logarithm that corresponds to . That is, given an equation of the form , it can be said that . Since is a special number, log to the base has its own name. That is, it is the natural logarithm and often called so is more often written as .

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

## Estimating Parameters

Consider the equation . Note that, according to BIDMAS, this is to the power of , then multiplied by . 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 and . 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 , it is possible to plot against . Recall that, in the equation , is the gradient and is the -intercept. In addition, we can write as and so, in the plot of against , is the gradient and is the -intercept.

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

It follows that can be written as and so, this time, in the plot of against , is the gradient and is the -intercept.

## Examples

The following table follows the relationship where the values are given to one decimal place. By plotting against , estimate the parameters and to 1 decimal place.

Solution:

Logs of the and values can be calculated as follows:

Note that log without a specified base is usually log base 10. Plotting against , we can see a straight line that has a gradient of  approximately 3.4 and a -intercept of approximately 0.4. It follows that and . Hence, the relationship between and as given by these points is approximately .

Sample Edexcel Exam Question 14:

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## Extra Resources

Graphing Logarithmic Functions