Which of the following functions grows the fastest as x goes to infinity? (4 points)

3^x
ln(x)
e^4x
x^10

The log function actually grows slower than any power of x. So we can throw option B out the window right away.

Exponential functions grow much much faster than powers of x. So we can throw away option D.

It's hard to compare exponentials which have different bases. We can apply a fancy little trick remembering that the log and exponential are inverse operations of one another.



We can use this algrebraic technique to rewrite the 3^x in terms of a base e.



Apply one of your exponent rules to bring the x down in front of the log,



This is easier to compare with e^(4x) now.

Which coefficient is larger? 4 or (ln3)?
It turns out the 4 is larger!

So e^(4x) will grow the fastest.

If that technique was too confusing...
then another approach would be to plug in a really really big x value like x=50 and see which function gives you the largest output.