### Video Transcript

The number of users of a new search engine is increasing every month and can be found by using the equation 𝑦 is equal to 500 times 1.19 raised to the power of 𝑥, where 𝑦 represents the number of users and 𝑥 represents the number of months since the search engine’s launch. If the search engine was launched on the 1st of March, in which month would the search engine have 2000 users?

We’re given an equation which represents the number of users of a search engine after 𝑥 months. We’re told 𝑦 is equal to 500 times 1.19 raised to the power of 𝑥, where 𝑦 is the number of users of the search engine and 𝑥 is the number of months since the search engine was launched. We’re told that the search engine was launched on the 1st of March. We need to use this information to determine in which month would the search engine have 2000 users.

The first thing we know is 𝑦 represents the number of users of our search engine. So for our search engine to have 2000 users, we must have that 𝑦 is equal to 2000, and we’re given an equation for 𝑦. So, we could substitute 𝑦 is equal to 2000 into this equation and then solve for 𝑥. 𝑥 would then give us the number of months after launch we would expect our search engine to have 2000 users. We need to solve the equation 2000 is equal to 500 times 1.19 raised to the power of 𝑥. To solve this equation for 𝑥, we’ll divide both sides of the equation through by 500. Since 2000 divided by 500 is equal to four, we get that four is equal to 1.19 raised to the power of 𝑥.

We can see this is an exponential equation for 𝑥. So, to solve this, we’re going to need to take the logs of both sides of the equation. We can do this by taking any logarithm base; for example, we could use logarithm base 1.19. And in fact, this would make it the easiest. However, we’re going to do this by taking the natural logarithm of both sides of the equation. This gives us the natural logarithm of four is equal to the natural logarithm of 1.19 raised to the power of 𝑥.

It’s also worth pointing out we can do this because we know both sides of our equation are positive. We still need to rearrange this for 𝑥. To do this, we’ll use the power rule for logarithms. Using the natural logarithm, this is the natural logarithm of 𝑎 to the power of 𝑏 is equal to 𝑏 times the natural logarithm of 𝑎. So, by applying this with our value of 𝑎 equal to 1.19 and our value of 𝑏 equal to 𝑥, we get the natural logarithm of four is equal to 𝑥 times the natural logarithm of 1.19.

Finally, we can solve for 𝑥 by dividing through by the natural logarithm of 1.19. This gives us that 𝑥 is equal to the natural logarithm of four divided by the natural logarithm of 1.19. And to two decimal places, this is approximately equal to 7.97. And remember, 𝑥 represents the number of months since the search engine’s launch. In other words, we’ve shown that our equation predicts the search engine will have 2000 users 7.97 months after the launch date of the 1st of March.

We might be tempted to round this number up to eight. However, this would not give us the answer that we want. 7.97 months after the 1st of March will be seven months after the 1st of March and then almost a full month. But it won’t be a full month. So, we just need to find seven months after the 1st of March. And seven months after March is October. It will be the end of October, but it will still be in the month of October.

In this question, we were given a real-world problem about the number of users of a new search engine after 𝑥 months. We were able to use an equation estimating the number of users of the search engine after 𝑥 months to approximate which month the search engine would have 2000 users. We got that this would happen in the month of October.