89. a scientist is studying the growth of a population of bacteria. at the beginning of her study, she has 800 bacteria. she notices that the population is quadrupling every hour.
c. find the time, in hours, when there will be 5,120,000 bacteria. express your answer as a logarithmic expression.

Step-by-step explanation:

The initial population of bacteria is 800 and we know that this number is quadrupling every hour.

We're going to find a function in terms of t (time) that gives us the population of bacteria at that time.

Since the population is quadrupling every hour the function in terms of t (where t is expressed in hours) is:

Now we need to find the time when there will be 5,120,000 bacterias. This means the time t when f(t) = 5,120,000

So we have 5,120,000 =

Therefore, the time when there will be 5,120,000 bacterias will be:

let's cover over the formula of the vertex:

y = a(x – h)^2 + k

in this formula, (h,k) are the vertex, always remember that the positive or negative value of the vertex is based on the sign it has.

for example, if the vertex is -3,-6 then both the x and y value would have the positive sign attached to it and if it were a positive x and y value, then a negative sign would be attached to it.

with that being said, for our problem the x value is a positive one so we are looking for a number with a negative sign next to it, in this case the first and second choice.

however, for the first answer choice it isn't quite part of the formula so we know that the answer choice correctly displays the x value.

to confirm, we can see if the -1 fits correctly with the second answer choice, and as mentioned above a negative number will have a positive sign attached to it. so, for our problem we can see that -1 has been changed to +1, so we can confirm that the second answer choice is correct.

hence, our final answer is y=(x - 1)^2 + 1