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?

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?

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