00:01
All right, this problem is an exponential growth equation and it's based off of the generic formula y equals ab to the x.
00:10
A is that initial amount where it starts, wherever you're talking about, right? x usually represents some kind of time.
00:18
So in our problem, it's talking about years.
00:21
And b, because it's going up 4 % and 5%, it's growing both of these, this is gonna be your growth rate.
00:29
So it's one plus whatever your rate is gonna be.
00:32
All right, so we have two cities.
00:36
We have anvil and brinker.
00:41
It says anvil has currently 22 ,000 people and it's growing at 4%.
00:51
Brinker starts with 1 ,000 people and it's growing at 5%.
00:59
So we'll clean these up.
01:03
What they wanna know is when will their populations be the same? because if you notice, brinker's a lot less, but it's growing faster, 5 % compared to 4%.
01:13
So all we have to do is set these two equations equal to each other.
01:17
And one plus 0 .04 is just 1 .04.
01:23
And one plus 0 .05 is 1 .05, all right? now to start, i'm just going to divide by 1 ,000 to make these numbers a little bit nicer.
01:34
And 22 ,000 divided by 1 ,000 is 22.
01:43
Now these are exponential functions.
01:45
And the only way to undo these exponential functions is to log each side.
01:50
Whatever you do to one, you have to do to the other.
01:53
But the thing that's interesting on the left -hand side here is we have two items being multiplied together.
02:00
We have 22 times 1 .04 to the x.
02:04
And the log properties tell us if i am multiplying those, i can add them separately.
02:17
And the other log property tells us when i have those exponents, that exponent can come down in front.
02:24
And that is an additional log property that we need in order to solve this problem...