The population of a city increases by four percent every year. How many years does it take for the population of the city to double?
This problem is an exponential growth problem. We’ve our starting population. And we need to multiply that by 𝑒 to the rate times time power. And that will give us 𝐴, our new amount. We don’t know the population of the city. But we do know we’re interested in it doubling.
We can actually use the number one as the value of our population. If one is the starting value of our population, then two would be the doubled value. We’re gonna multiply that by 𝑒 to the 0.04 power. 0.04 is four percent written as a decimal. And our missing value is 𝑡. We want to know how many years it would take to go from one to two in our population. And that means we need to isolate 𝑡. We need to get 𝑡 by itself. Anything multiplied by one is itself.
To isolate 𝑡, we need to get it out of the exponent. And to do that, we’ll need to take the natural log of 𝑒 to the 0.04𝑡 power. And if we take the natural log on the right side of the equation, we need to take the natural log on the left side of the equation. The natural log of two will be equal to 0.04 times 𝑡. This is because the natural log of 𝑒 to any power equals whatever is in the exponent.
Now that we know that and our goal is to isolate 𝑡, we can divide both sides of the equation by the rate 0.04. And 𝑡 equals 17.328 continuing. So it’s just over 17 years, 17.3 years. But in this kind of question, we can’t put 17.3 years. We need to round to the nearest year. But if we rounded down to 17, the population would it be doubled all the way? Year 18 would be the first year that the population has doubled.
So we say that it takes 18 full years for the population of the city to double.