Investigation

## 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?

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 $x$?