(I skipped the lengthy work problem background as it is irrelevant)
The number of bacteria needed to break down an oil spill is 1000000 per millitre of oil. The bacteria double in number every two days.The starting concentration of bacteria is 1000 bacteria per millimetre.The situation can be modelled by the equation C=1000(2^d) where C is estimated concentration of bacteria and d is the number of 2-day periods the bacteria grow.Approximately how long would it take for the bacteria to reach the required concentartion?
C= 1000(2^d)
1000000=1000(2^d)
1000=2^d
The only part that confuses me is this: how do I go on from there?
take the log to any base of both sides
for example base 10
log 1000 = d log 2
d = 3/.301 = 9.96
2 day periods
or 20 days
check my reply to your earlier efficiency problem
O.o We haven't even come close to logarithms. What grade level is this?
hmmm
well, you need 2^something = 1000
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = about 1000 so d = about 10
so 2*10 = 20 days. It said approximate :)
These problems are pretty advanced for class that has not reached logarithms.
This was one of the advanced problems in my textbook. I think the author intended to solve the problem using the second method.
Thank you,once again for helping me!
You are welcome :)
To solve the equation 1000 = 2^d, you need to take the logarithm of both sides. In this case, the logarithm used is the base-2 logarithm because the base of the exponent in the equation is 2.
The base-2 logarithm of a number x is denoted as log₂(x) or simply log(x) when the base is assumed to be 2.
So, applying the base-2 logarithm to both sides of the equation, we get:
log₂(1000) = log₂(2^d)
Using the logarithm property log₂(x^a) = a * log₂(x), the equation becomes:
log₂(1000) = d * log₂(2)
Since log₂(2) = 1, we can simplify further:
log₂(1000) = d
Now, to find the approximate value of d, you can use a calculator to evaluate log₂(1000). The answer will give you d, which represents the number of 2-day periods it would take for the bacteria to reach the required concentration.
Remember that the logarithm gives you an exponent, so d will not represent the number of days directly. Since each period corresponds to two days, you can multiply d by 2 to get an approximate number of days.