Mathematical Investigations III

Name:

Logarithms

Review

1. The value of is

2. If , then m =

3. If and a > 1, then which of the following statements is (are) correct?

a. If x = 1, then y = 0.

b. If x = a, then y = 1.

c. If x = , then y is imaginary.

d. If 0 < x £ 1, then y £ 0 and decreases without limit as x approaches 0.

4. Solve for z:

5. a. b.

c. = d. =

6. If you are given = t and = r, give the values of each of the following without using either tables or a calculator. [If this cannot be done, write "Impossible"]

a. = b. =

c. = d. =

7. Eliminate the logs to solve for p in terms of q: log(p) + log(q) = log(p + q).

8. Simplify =

9. Consider the graphs of y = 2 log(x) and y = log(2x). Without using your graphing calculator, which of the following is(are) true?

a. They do not intersect. b. They intersect in only 1 point.

c. They intersect in only two points. d. They coincide.

e. They intersect in a finite number of points, but more than twice.

10. Solve for n in terms of s, a, and r:

11. Solve each for x:

a. b.

12. If , then =

13. Write the following expression as a single log:

14. Solve for x:

15. Simplify: =

16. Solve for x:

17. Solve for x:

18. Write as a log of a single expression:

19. To do a bone scan in the hospital, the patient is injected with a radioactive substance with a half-life of approximately 6 hours. There is a calcium additive to make the substance adhere to the bones, which then become visible when pictures are taken. After how many hours will the radioactive substance decay to 10% of the original amount?

20. Kinshasa, the capital of the Congo, had a metropolitan area population of approximately 4 million people in 1993. It is estimated that the population is growing at a rate of almost 10% per year. Assuming this rate, how long did it take the population to double? How many years did it take for the population to reach 20 million?

21. The half-life of radioactive radium is approximately 1600 years.

a. If a sample initially contains 20 grams of radium, how much radioactive will be left after 700 years?

b. After how many years will 60% of a sample have decayed?

22. Solve for x: (Give exact answers whenever possible.)

a. b.

c. d.

23. Given the graphs of any of the following, you should be able to write the equation (See problem 25). Here, you are given the equation. Graph the functions without using your calculator.

a. b.

c. d.

e. f.

g. h.

24. Prove each of the following:

a. Given log(ab) = log(a) + log(b), b. Given: ,

derive derive:

25. For each of the following, find a function of the form :

Logs 14.XXX Rev. S12