Question

(Alg 2) A culture of 10 bacteria is started, and the number of bacteria will double every hour. In about how many hours will there be 3,000,000 bacteria. Round to the nearest tenths.

Answer 1

Lets say it doubled n times.
Then 3000000 = 10 * 2 * 2 * 2 * ... * 2 where the multiplication is repeated n times
So we have 3000000 = 10*(2)^n. Now you can solve for n

Answer 2

@Xavier I'm trying to figure out how to solve for n, I'm looking all in my notes and i can't find anything. Found you have 3000000 = 20^n or how would you do it?

Answer 3

I can do some of the other chemistry problems but this one is stumping me!

Answer 4

Be careful. You can't say it's 20^n. That's something that started at 1 and multiplied by 20 every hour. 20^n is different that 10(2)^n = 5(2)^(n+1)
3000000 = 10*(2)^n
300000 = 2^n
Then take the logarithm of both sides base 2 and solve from there.

Answer 5

Okay! I just attempted it and got 3.2 hours? @Xavier

Answer 6

How did you get that? What expression did you type into a calculator?

Answer 7

You should also check the practicality of your answer. 3.2 hours means less than 4 doublings.
Doubling 10 four times: 10 20 40 80 160 so not nearly enough

Answer 8

I got 6.477121255 for the log of 3000000 and then divided it by 2. I think I'm confusing this with another type of equation my teacher made us do. @Xavier

Answer 9

Yes the first step you want to do is take the logarithm. The two dissapears
3000000 = 10(2)^n
300000 = 2^n
\[\log_{2} 300000 = \log_{2} 2^{n} = n\]

Answer 10

@dellzasaur Your mistake was that you did \[\log_{10} 3000000 = 6.477121255\] not \[\log_2 3000000\]

Answer 11

You can do the problem just fine with that, you just need to use the change of base formula.
\[\log_a x = \frac{\log_b x}{\log_b a}\]

Answer 12

Divide your value for the log of 3000000 by the log of 2...

Also, you want to divide both sides by 10 before you take the log, so it is really the log of 300000 you want to find, not 3000000

Finally, as a quick sanity check, \[2^{10} = 1024 \implies 2^{20} = 1024*1024 \approx 1,000,000\]After 20 doublings, you would have 10,000,000 bacteria.

Answer 13

@whpalmer4 Thank you so much!! That helps a lot.

Thank you both so much for your help!! :) @Xavier I appreciate it!