EVALUATING LOGARITHMIC EXPRESSIONS WORKSHEET

Evaluate each of the following logarithmic expressions :

Problem 1 :

log₂ 24 - log₂ 3

Problem 2 :

log₃ 27 + log₃ 729

Problem 3 :

log₁₀ 8 + log₁₀ 5 - log₁₀ 4

Problem 4 :

log₇ 98 - log₇ 14 + log₇ 343

Problem 5 :

Problem 6 :

log 50 - (log 25 + log 2)

Problem 7 :

log₂ 2 + log₂ 3 - log₂ 6 - log₂ 8

Problem 8 :

4log₂ 2 + log₃ 27 - 3log₂ 2 - log₂ 81

Problem 9 :

log₃ 2 ⋅ log₄ 3 ⋅ log₅ 4 ⋅ log₆ 5 ⋅ log₇ 6 ⋅ log₈ 7

Problem 10 :

log₇ 21 + log₇ 77 + log₇ 88 - log₇ 121 - log₇ 24

Problem 11 :

Problem 12 :

5log₁₀ 2 + 2log₁₀ 3 - 6log₆₄ 4

Problem 13 :

Answers

1. Answer :

= log₂ 24 - log₂ 3

= log₂ 8

= log₂ 2³

= 3log₂ 2

= 3(1)

= 3

2. Answer :

= log₃ 27 + log₃ 729

= log₃ 3³ + log₃ 3⁶

= 3log₃ 3 + 6log₃ 3

= 3(1) + 6(1)

= 3 + 6

= 9

3. Answer :

= log₁₀ 8 + log₁₀ 5 - log₁₀ 4

= log₁₀ (8 ⋅ 5) - log₁₀ 4

= = log₁₀ 40 - log₁₀ 4

= log₁₀ 10

= 1

4`. Answer :

= log₇ 98 - log₇ 14 + log₇ 343

= log₇ 7 + log₇ 7³

= 1 + 3log₇ 7

= 1 + 3(1)

= 1 + 3

= 4

5. Answer :

6. Answer :

= log 50 - (log 25 + log 2)

= log 50 - log (25 ⋅ 2)

= log 50 - log 50

= 0

7. Answer :

= log₂ 2 + log₂ 3 - log₂ 6 - log₂ 8

= (log₂ 2 + log₂ 3) - (log₂ 6 + log₂ 8)

= log₂ (2 ⋅ 3) - log₂ (6 ⋅ 8)

= log₂ 6 - log₂ 48

= log₂ 2^-3

= -3log₂ 2

= -3(1)

= -3

8. Answer :

= 4log₂ 2 + log₃ 27 - 3log₂ 2 - log₃ 81

= 4(1) + log₃ 3³ - 3(1) - log₃ 3⁴

= 4 + 3log₃ 3 - 3 - 4log₃ 3

= 4 + 3(1) - 3 - 4(1)

= 4 + 3 - 3 - 4

= 0

9. Answer :

= log₃2 ⋅ log₄3 ⋅ log₅4 ⋅ log₆5 ⋅ log₇6 ⋅ log₈7

In the above expression, logarithms have differemnt bases. Group the logarithms with the same base and simplify.

= (log₃ 2 ⋅ log₄ 3) ⋅ (log₅ 4 ⋅ log₆ 5) ⋅  (log₇ 6 ⋅ log₈ 7)

= log₄ 2 ⋅ log₆ 4 ⋅ log₈ 6

= log₆ 2 ⋅ log₈ 6

= log₈ 2

10. Answer :

= log₇ 21 + log₇ 77 + log₇ 88 - log₇ 121 - log₇ 24

= log₇ (21 ⋅ 77 ⋅ 88) - (log₇ 121 + log₇ 24)

= log₇ (21 ⋅ 77 ⋅ 88) - log₇ (121 ⋅ 24)

= log₇ 142296 - log₇ 2904

= log₇ 49

= log₇ 7²

= 2log₇ 7

= 2(1)

= 2

11. Answer :

= log₈ 64

= log₈ 8²

= 2log₈ 8

= 2(1)

= 2

12. Answer :

= 5log₁₀ 2 + 2log₁₀ 3 - 6log₆₄ 4

= 5log₁₀ 2 + 2log₁₀ 3 - (2 ⋅ 3)log₆₄ 4

= log₁₀ 2⁵ + log₁₀ 3² - 2log₆₄ 4³

= log₁₀ 32 + log₁₀ 9 - 2log₆₄ 64

= log₁₀ 32 + log₁₀ 9 - 2(1)

= log₁₀ 32 + log₁₀ 9 - 2log₁₀ 10

= log₁₀ (32 ⋅ 9) - log₁₀ 10²

= log₁₀ 288 - log₁₀ 100

13. Answer :

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