Things you might have thought about

Which is bigger:

  • \(2^x\) or \(x^2\)?

The first is an exponential expression and the second is a quadratic.

In the special case when \(x=0\), we find the exponential is bigger, \(2^0>0^2\). Try some other values of \(x\).

Are there any values of \(x\) for which \(2^x\) and \(x^2\) are the same?

Which expression grows faster when \(x\) gets large?

What about when \(x\) is negative?

You might think about the expressions as functions of \(x\) and what their graphs look like. When will the two graphs intersect and what does this tell us?

You could make a table of values to get a feel for how the functions behave and which is bigger.

Which is bigger:

  • \(a^x\) or \(x^a\)?

In this more general case, we can change the value of \(a\) as well as \(x\). The second expression is no longer a quadratic but a power of \(x\).

The case \(a=2\) is the same as we looked at above. What about when \(a=3\)?

If we think of them as functions of \(x\), how does the rate of growth of each function change as \(a\) changes?

When are the values of the two expressions the same?

Thinking about the graphs of \(y=a^x\) and \(y=x^a\), what happens as \(a\) gets small, or negative? Does \(a\) have to be a whole number?

Can you write down, in terms of \(a\), the coordinates of one of the graph intersections? How many others are there?

The cases \(a=0\) and \(a=1\) are special. Can you describe them? Can you find any other special cases?

You may find it helpful to use graphing software such as Desmos or GeoGebra.

Is there a general rule about which of the expressions is bigger for large values of \(x\)?