### Instruction: The bacteria in a lab dish double in number every four hours. If 500 bacteria cells are in the dish now, in how many hours will there be exactly 32,000 bacteria?### Response: I need to find a way to relate the current number of bacteria to the number of bacteria after some time.
I know that the bacteria double every four hours, so that means they multiply by 2 every time the time increases by 4 hours.
So, if I let x be the number of hours that pass, and y be the number of bacteria after x hours, I can write an equation of the form y = 2^(x/4) * 500.
This equation says that the number of bacteria y is equal to 500 times 2 raised to the power of x/4, which is the number of four-hour intervals that have passed.
To check if this equation makes sense, I can plug in x = 0 and see if I get y = 500, which is the initial number of bacteria.
If I do that, I get y = 2^(0/4) * 500, which is 2^0 * 500, which is 1 * 500, which is 500.
So that works.
Now, I want to find the value of x that makes y equal to 32,000, which is the final number of bacteria.
So I set y = 32,000 and solve for x.
I get 32,000 = 2^(x/4) * 500.
I can divide both sides by 500 to get 64 = 2^(x/4).
To get rid of the exponent, I can take the logarithm of both sides, using base 2.
I get log_2(64) = log_2(2^(x/4)).
Using the logarithm property that log_b(b^a) = a, I can simplify the right-hand side to get x/4 = log_2(64).
To find the value of log_2(64), I can think of what power of 2 gives me 64.
I know that 2^6 = 64, so log_2(64) = 6.
Plugging that in, I get x/4 = 6.
Multiplying both sides by 4, I get x = 24.
So, it takes 24 hours for the bacteria to grow from 500 to 32,000.