SOLUTION: If all who hear a rumor repeat it to two people a day, and if 20 people start the rumor, then number of people N who have heard the rumor after t days is given by N(t) = 20(3)t.

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: If all who hear a rumor repeat it to two people a day, and if 20 people start the rumor, then number of people N who have heard the rumor after t days is given by N(t) = 20(3)t.       Log On


   

Question 225276: If all who hear a rumor repeat it to two people a day, and if 20 people start the rumor, then number of people N who have heard the rumor after t days is given by N(t) = 20(3)t. After what amount of time will 1000 have heard the rumor?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
N%28t%29+=+20%283%29%5Et
You're given n(t) =1000 and you've been asked to find t:
1000++=+20%283%29%5Et
So solve for t we're going to "peel away" the rest of the right side (i.e. the 20 and the 3). We'll start by dividing both sides by 20:
50+=+3%5Et
Next, to get rid of the 3, we will need to use logarithms. If 50 and 3%5Et are equal then their logarithms are equal, too. (You should use a base for the logarithms that your calculator can handle. Usually base 10 or base e (aka natural) logarithms are best.):
log%28%2850%29%29+=+log%28%283%5Et%29%29
Now we can use the property of logarithms, log%28a%2C+%28b%5Ec%29%29+=+c%2Alog%28a%2C+%28b%29%29, to "move" the t out of the exponent:
log%28%2850%29%29+=+t%2Alog%28%283%29%29
We can now solve for t by dividing both sides by log%28%283%29%29:
log%28%2850%29%29%2Flog%28%283%29%29+=+t
For our final answer we can use our calculator on the two logarithms on the left side:
1.6989700043360188047862611052755%2F0.47712125471966243729502790325512+=+t
3.5608767950073117714936079297+=+t
Of course you can round off these decimals as you prefer.