78 CHAPTER 2. FUNCTIONS

13. Find all solutions to log2 (x+4) = log4 (x+16) .

14. If the interest rate is 4% compounded continuously, how long does it take a givenamount of money to double? This means the rate per year is .04.

15. If the interest rate is 6% compounded continuously, how long does it take a givenamount of money to double? This means the rate per year is .06.

16. The population of bacteria grows exponentially. It is observed that every hour thispopulation doubles. How long will it take to have eight times as many bacteria as atthe beginning?

17. A pesticide has a half life of 27 years. How long will it take to have only 1/4 theinitial amount?

18. Suppose 5% interest is compounded continuously and you make a payment of $100at the end of every year, starting with an initial $1000. How much will you have atthe end of 10 years?

19. Measurements are taken of an exponentially decaying substance and it is found thatafter 5 years there is .9 of the amount which was present at the start. What is the halflife of this substance?

20. Consider the logistic equation y = K1+e−rtCK where y(0) = 20 and K = 30,r = 1. Find

C in the logistic equation to conform with this information. Then graph the resultingfunction.

21. In the logistic equation, explain why for large t it will always be close to K. In otherwords, explain why it has a horizontal asymptote of the form y = K.

22. Suppose f : R→ R is any function. Let

h(t)≡ 12( f (t)+ f (−t)) , g(t) =

12( f (t)− f (−t)) .

23. Solve for x. 3log27(x) = 4.

2.10 Videoscircular functions logs and exponentials