SOLUTION: Assume that the number of viruses present in a sample is modeled by the exponential function "f(t) = 10t," where t is the elapsed time in minutes. How would you apply logarithms to

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Assume that the number of viruses present in a sample is modeled by the exponential function "f(t) = 10t," where t is the elapsed time in minutes. How would you apply logarithms to      Log On


   

Question 270128: Assume that the number of viruses present in a sample is modeled by the exponential function "f(t) = 10t," where t is the elapsed time in minutes. How would you apply logarithms to determine when the sample will grow to 5 billion viruses?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
(Please use "^" to indicate an exponent. IOW: f(t) = 10^t)

f%28t%29+=+10%5Et
with t as the time (in minutes) and f(t) is the number of viruses at time t. So to find when the number the viruses reaches 5 billion we replace f(t) with 5 billion and solve for t:
5000000000+=+10%5Et
Solving for t will require that we get it out of the exponent. This is why we use logarithms. So we start by finding the logarithm of each side. You can use any base of logarithm but it will be easier if we use a base your calculator "knows" (like base 10 or base e (aka ln)). And since the equation has 10 to a power, it will be easiest if we use base 10. So that is what we will use:
log%28%285000000000%29%29+=+log%28%2810%5Et%29%29
There is a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, which allows us to move an exponent of the argument out in front. (It is this property that is the reason we use logarithms to get variables out of exponents.) Using this property on your equation we get:
log%28%285000000000%29%29+=+t%2Alog%28%2810%29%29
Since log(10) = 1 this simplifies to
log%28%285000000000%29%29+=+t
This is an exact answer for the time when the number of viruses reaches 5 billion. If you want a decimal approximation for the answer, then get out your calculator and use it to find log(5000000000).