Logarithms



Logarithms

I. Convert to an exponential equation:

1. t = log4 7 2. log5 5 = 1 3. log10 7 = 0.845

4. loga 10 = 2.3036 5. loga 0.38 = –0.9676 6. loga W = –w

7. h = log6 29 8. log10 0.1 = –1 9. log10 3 = 0.4771

10. logt Q = k 11. loga 0.906 = –0.0987 12. log2 32 = 5

13. log10 0.01 = –2 14. logm P = a 15. logr M = –x

II. Solve:

1. log10 x = 3 2. log5 = x 3. logx 16 = 2

4. log3 x = –2 5. log2 16 = x 6. log3 x = 2

7. logx 64 = 3 8. log8 x = 9. log3 3 = x

10. log4 x = 3 11. log2 x = –1 12. log32 x =

13. |log3 x| = 3 14. logx = –3 15. log8 (2x – 3) = –1

16. log125 x = 17. logb b2x2 = x 18. logx =

19. log( (4 = x 20. log4 (3x – 2) = 2 21. log9 (x2 + 2x) =

III. Simplify:

1. log 2. log2 (log2 256) 3. log

4. log4 (log3 81) 5. log25

IV. Expand the following:

1. logbPQ 2. logb 3. loga

4. loga 5. logm 6. loga

7. loga 8. logb 9. loga

10. loga 11. ln 12. ln

V. Express as a single logarithm:

1. logaC + logaA + logaB + logaI + logaN 2. log2x – log225

3. 5 loga x – loga y + loga z 4. loga x – 7 loga y + loga z

5. loga – loga 6. loga + loga

7. loga x – loga y 8. loga x + 4 loga y – 3 loga x

9. loga 2x + 3(loga x – loga y) 10. loga x2 – 2 loga

11. loga – loga 12. loga (x2 – 4) – loga (x – 2)

13. [2 ln(x + 1) + ln x – ln(x2 – 1)] 14. 2[ln(x + 1) + ln(x – 1)] – 3 ln(x2 – 1)

VI. Use the change of base formula to rewrite the given logarithm in the indicated base.

1. log3 5 in base 10 2. log5 3 in base e

3. log2 x in base 10 4. logx y in base y

5. log4 8 in base 2 6. log10 in base 5

VII. Evaluate the following given that logb 2 ( 0.3562, logb 3 ( 0.5646, and logb 5 ( 0.8271.

1. logb 6 2. logb 3. logb 25

4. logb 5. logb 6. logb

7. logb 8. logb 15 9. logb

10. logb 18 11. logb 12. logb

13. logb (3b2) 14. logb 1

Answers

I.

1. 4t = 7 2. 51 = 5 3. 100.845 = 7

4. a2.3036 = 10 5. a–0.9676 = 0.38 6. a–w = W

7. 6h = 29 8. 10–1 = 0.1 9. 100.4771 = 3

10. tk = Q 11. a–0.0987 = 0.906 12. 25 = 32

13. 10–2 = 0.01 14. ma = P 15. r–x = M

II.

1. 1000 2. –2 3. 4 4.

5. 4 6. 9 7. 4 8. 2

9. 1 10. 64 11. 12. 2

13. 27, 14. 16. 25 17. 0,

18. 362 19. 4 20. 6 21. 1, –3

III.

1. 3 2. 3 3. –8 4. 1 5. –2

IV.

1. logb P + logb Q 2. logb P – logb Q 3. [3 loga z – loga x – loga y]

4. 2 loga x – 3 loga y – 1 5. 3 logm a + 4 logm b – 9 logm n – 5

6. 3 loga p + 2 loga q – 4 loga z 7. [loga b – 3 loga c + 1]

8. logb a – 3 logb m – 4 logb n + 5 9. [ loga (2 + x) + loga (2 – x)]

10. – loga (x + y) + loga (x – y) 11. ln(x + 1) + ln(x – 1) – 3 ln x

12. ln x + ln(x + 2)

V.

1. logaCABIN 2. log2 3. loga

4. loga 5. – loga b 6. loga b

7. loga 8. loga 9. loga

10. loga x 11. loga or – loga x

12. loga (x + 2) 13. ln 14. – ln(x2 – 1) or ln

VI.

1. 2. 3.

4. 5. 6. – 1 – log5 2

VII.

1. 0.9208 2. 0.2084 3. 1.6542

4. 0.1781 5. –0.7124 6. 0.9136

7. 2.0367 8. 1.3917 9. 0.2625

10. 1.4854 11. 0.7730 12. 0.7396

13. 2.5646 14. 0

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