Exponential Graphs

Exponential graphs are those of the form y=a^x for positive a. You can sketch the graph of y=a^x 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 a^0=1 for any a.

The following table shows coordinates for the graph y=2^x 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

exponential graphs

It follows that the graph of y=a^x for a\textgreater1 will have a shape like y=2^x.


The graph of y=a^x for 0\textless a\textless 1 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 y=0.5^x 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

exponential graphs

For any number 0\textless a\textless 1, the graph will have the same shape as y=0.5^x.

For a=1, the graph of y=a^x is the horizontal line y=1. 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.


exponential graphs

The diagram shows the graph of y=e^{x} 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

exponential graphsAs well as exponential graphs, there are logarithmic graphs. \log_a(x) is considered to be the inverse of a^x – see more on logs. It follows that \log_a(x) (blue solid line) is the inverse of a^x (red dotted line) and so their graphs are reflections of each other in the line y=x (green dotted line). Since \log_a(x) and a^x are mathematical inverses we have that \log_a\left(a^x\right)=a^{\log_a(x)}=x.
Specifically, the natural logarithm is the logarithm that corresponds to e. That is, given an equation of the form y=e^{x}, it can be said that x=\log_e(y). 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 x=\log_e(y) is more often written as x=\ln(y).

\ln(x) and e^x are mathematical inverses and we have that \ln\left(e^x\right)=e^{\ln(x)}=x. Notice that log_a for any positive a (including ln) cannot be evaluated for negative x – see more on logs.

YouTubeGraphing Logarithmic Functions


Estimating Parameters for y=ax^n

Consider the equation y=ax^n. 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:


\begin{array}{c}\log(y)=\log(ax^n)\\\Longrightarrow \log(y)=\log(a)+\log(x^n)\\\Longrightarrow \log(y)=\log(a)+n\log(x)\end{array}

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 y=ax^n where the y values are given to one decimal place. By plotting \log(y) against \log(x), 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 y=kb^x

Now consider the equation y=kb^x. 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:

\begin{array}{l}\log(y)=\log(kb^x)\\\Longrightarrow \log(y)=\log(k)+\log(b^x)\\\Longrightarrow \log(y)=\log(k)+x\log(b)\end{array}

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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