SOLUTION: The population growth of an animal species is described by F(t) = 700 + 50 log3 (2t + 1), where t is measured in months. Find the population of this species in an area 13 month(s

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: The population growth of an animal species is described by F(t) = 700 + 50 log3 (2t + 1), where t is measured in months. Find the population of this species in an area 13 month(s      Log On


   

Question 250805: The population growth of an animal species is described by
F(t) = 700 + 50 log3 (2t + 1),
where t is measured in months. Find the population of this species in an area 13 month(s) after the species is introduced.
Thank you for any help you can offer.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
So we have a value for t and we've been asked to find the value of F(t):
F%2813%29+=+700+%2B+50%2Alog%283%2C+%282%2813%29+%2B+1%29%29
Simplifying we get:
F%2813%29+=+700+%2B+50%2Alog%283%2C+%2826+%2B+1%29%29
F%2813%29+=+700+%2B+50%2Alog%283%2C+%2827%29%29
At this point we want to find the base 3 logarithm of 27. I don't know of a calculator that can find base 3 logarithms. So what do we do?
  • A logarithm can be found "by hand" if the argument is a power of the base of the logarithm.
  • We can use the base conversion formula for logarithms: log%28a%2C+%28q%29%29+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29 This gives us a way to convert a logarithm from one base, "a", into an expression of another base, "b".
  • If you can't remember the base conversion formula, like me, then you can derive it:
    • Use a temporary variable. Let p+=+log%28a%2C+%28q%29%29
    • Rewrite this in exponential form:
      a%5Ep+=+q
    • Find the base b logarithm of both sides:
      log%28b%2C+%28a%5Ep%29%29+=+log%28b%2C+%28q%29%29
    • Use the property of logairhtms, log%28b%2C+%28x%5Ey%29%29+=+y%2Alog%28b%2C+%28x%29%29, to move the exponent in the argument out front:
      p%2Alog%28b%2C+%28a%29%29+=+log%28b%2C+%28q%29%29
    • Divide both sides by log%28b%2C+%28a%29%29:
      p+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29
    • Replace the temporary variable with the expression it represents:
      log%28a%2C+%28q%29%29+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29

With our problem the argument of our logarithm, 27, is a power of the base, 3. 27+=+3%5E3. So log%283%2C+%2827%29%29+=+3. If we do not recognize that 27 is a power of 3 then we must use the base conversion formula to convert the base into one our calculators can find:
log%283%2C+%2827%29%29+=+log%28%2827%29%29%2Flog%28%283%29%29
or
log%283%2C+%2827%29%29+=+ln%2827%29%2Fln%283%29
Correctly using either of these with our calculator should give us 3 (or a decimal very, very close to 3).

Now that we have figured out that log%283%2C+%2827%29%29+=+3, our equation becomes:
F%2813%29+=+700+%2B+50%2A3
which simplifies to
F%2813%29+=+700+%2B+150
F%2813%29+=+850