00:01
In this example, the function f of x equals the natural log x has been provided, and our goal here is to compare that function to the functions provided in this set to specifically determine which ones have the same, slower, or faster rate of growth.
00:18
Let's start off with the first function in this set, log base 3 of x.
00:23
If we want to determine how this grows relative to the natural log x, we can always take the limit as x goes to infinity, to compare.
00:32
Let's place, say, the given function in the set, log base 3 of x in the numerator, and the function we're comparing everything to, natural log x, in the denominator.
00:44
Now, as x goes to infinity, this is an indeterminate form of type infinity over infinity, so we can use lopital's rule to evaluate the limit.
00:53
We'll have the limit as x goes to infinity, and for the numerator, the derivative of log base 3 of x is 1 over x times the natural log of 3.
01:05
Then the derivative of the natural log of x is 1 over x, and if we simplify this, we'll have altogether just 1 over the natural log of 3 since x cancels out.
01:21
Then we see that we have a constant when we compared the 2, but what this means is the growth rate is the same and y equals log base 3 of x will go into this category.
01:34
When we go to the next function for natural log 2x, natural log square root of x, notice that natural log 2x can be rewritten as log base 2 plus the natural log of x.
01:48
This was log base e of 2 or natural log of 2 rather.
01:52
Let's write in natural log 2 here.
01:58
So this function is essentially the same as our given function...