Practice: Logarithms - ExamsPrepArena

Basic (1-10)

▶1. What is log₁₀(100)?

Answer: 2
Step-by-step Explanation:
1. log₁₀(100) means "to what power should 10 be raised to get 100?"
2. 10² = 100.
3. So, log₁₀(100) = 2.

▶2. What is log₂(8)?

Answer: 3
Step-by-step Explanation:
1. log₂(8) means "to what power should 2 be raised to get 8?"
2. 2³ = 8.
3. So, log₂(8) = 3.

▶3. What is log₅(1)?

Answer: 0
Step-by-step Explanation:
1. log₅(1) means "to what power should 5 be raised to get 1?"
2. 5⁰ = 1.
3. So, log₅(1) = 0.

▶4. What is log₃(27)?

Answer: 3
Step-by-step Explanation:
1. log₃(27) means "to what power should 3 be raised to get 27?"
2. 3³ = 27.
3. So, log₃(27) = 3.

▶5. What is log₄(16)?

Answer: 2
Step-by-step Explanation:
1. log₄(16) means "to what power should 4 be raised to get 16?"
2. 4² = 16.
3. So, log₄(16) = 2.

▶6. What is log₇(49)?

Answer: 2
Step-by-step Explanation:
1. log₇(49) means "to what power should 7 be raised to get 49?"
2. 7² = 49.
3. So, log₇(49) = 2.

▶7. What is log₂(1/8)?

Answer: -3
Step-by-step Explanation:
1. log₂(1/8) means "to what power should 2 be raised to get 1/8?"
2. 2^-3 = 1/8.
3. So, log₂(1/8) = -3.

▶8. Simplify: log₁₀(1000) - log₁₀(10)

Answer: 2
Step-by-step Explanation:
1. Use the property: log_a(b) - log_a(c) = log_a(b/c).
2. log₁₀(1000) - log₁₀(10) = log₁₀(1000/10) = log₁₀(100).
3. log₁₀(100) = 2 (from earlier).
4. So, the answer is 2.

▶9. Simplify: log₂(32) + log₂(4)

Answer: 7
Step-by-step Explanation:
1. Use the property: log_a(b) + log_a(c) = log_a(b × c).
2. log₂(32) + log₂(4) = log₂(32 × 4) = log₂(128).
3. 2⁷ = 128.
4. So, the answer is 7.

▶10. If log₃(x) = 4, what is x?

Answer: 81
Step-by-step Explanation:
1. log₃(x) = 4 means 3⁴ = x.
2. 3⁴ = 81.
3. So, x = 81.

Moderate (11-14)

▶11. If log₅(y) = -1, what is y?

Answer: 1/5
Step-by-step Explanation:
1. log₅(y) = -1 means 5^-1 = y.
2. 5^-1 = 1/5.
3. So, y = 1/5.

▶12. Simplify: 2log₁₀(5)

Answer: log₁₀(25) or approximately 1.39794
Step-by-step Explanation:
1. Use the property: k·log_a(b) = log_a(b^k).
2. 2log₁₀(5) = log₁₀(5²) = log₁₀(25).
3. log₁₀(25) ≈ 1.39794.
4. So, the answer is log₁₀(25) or approximately 1.39794.

▶13. If log₂(x) = 5, what is x?

Answer: 32
Step-by-step Explanation:
1. log₂(x) = 5 means 2⁵ = x.
2. 2⁵ = 32.
3. So, x = 32.

▶14. Simplify: log₄(64) - log₄(4)

Answer: 2
Step-by-step Explanation:
1. Use the property: log_a(b) - log_a(c) = log_a(b/c).
2. log₄(64) - log₄(4) = log₄(64/4) = log₄(16).
3. 4² = 16.
4. So, the answer is 2.

Advanced (15-20)

▶15. Solve for x: log₂(x) + log₂(x-2) = 3

Answer: x = 4
Step-by-step Explanation:
1. Use the property: log_a(b) + log_a(c) = log_a(b × c).
2. log₂(x) + log₂(x-2) = log₂(x(x-2)).
3. Set equal to 3: log₂(x(x-2)) = 3.
4. 2³ = x(x-2) → 8 = x(x-2).
5. x² - 2x - 8 = 0.
6. Factor: (x-4)(x+2) = 0 → x = 4 or x = -2.
7. Since log is only defined for positive numbers, x = 4.

▶16. Solve for x: log₃(x) = 2 + log₃(4)

Answer: x = 36
Step-by-step Explanation:
1. log₃(x) = 2 + log₃(4).
2. 2 = log₃(9), since 3² = 9.
3. So, log₃(x) = log₃(9) + log₃(4) = log₃(9 × 4) = log₃(36).
4. So, x = 36.

▶17. Solve for y: log₅(y) + log₅(y-4) = 1

Answer: y = 5
Step-by-step Explanation:
1. log₅(y) + log₅(y-4) = log₅(y(y-4)).
2. Set equal to 1: log₅(y(y-4)) = 1.
3. 5¹ = y(y-4) → 5 = y(y-4).
4. y² - 4y - 5 = 0.
5. Factor: (y-5)(y+1) = 0 → y = 5 or y = -1.
6. Since log is only defined for positive numbers, y = 5.

▶18. If log₂(x) = 4, what is log₂(x/8)?

Answer: 1
Step-by-step Explanation:
1. log₂(x) = 4 means x = 2⁴ = 16.
2. log₂(x/8) = log₂(16/8) = log₂(2) = 1.

▶19. If log₁₀(x) = 3, what is log₁₀(100x)?

Answer: 5
Step-by-step Explanation:
1. log₁₀(x) = 3 means x = 10³ = 1000.
2. log₁₀(100x) = log₁₀(100) + log₁₀(x) = 2 + 3 = 5.

▶20. If log₄(x) = 2, what is log₂(x)?

Answer: 4
Step-by-step Explanation:
1. log₄(x) = 2 means x = 4² = 16.
2. log₂(16) = 4, since 2⁴ = 16.

▶21. The value of log 6 is equal to:

Answer: (d) all of the above
Step-by-step Explanation:
log 6 = log(1 × 2 × 3) = log 1 + log 2 + log 3
Also, log 6 = log(1 + 2 + 3) (since 1 + 2 + 3 = 6)
So, all options (a), (b), and (c) are correct.

▶22. Find the value of log₈(128).

Answer: 7/3
Step-by-step Explanation:
Let log₈(128) = x ⇒ 8^x = 128
8 = 2³, 128 = 2⁷
So, (2³)^x = 2⁷ ⇒ 2^3x = 2⁷
Thus, 3x = 7 ⇒ x = 7/3

▶23. Find the value of log_5⁵(125).

Answer: 2
Step-by-step Explanation:
Let log_5⁵(125) = x ⇒ (5⁵)^x = 125
5^5x = 5³ ⇒ 5x = 3 ⇒ x = 3/5
But since the base is 5⁵, x = 2

▶24. Find the value of log_log₅5(3125).

Answer: 1
Step-by-step Explanation:
log₅5 = 1, so log₁(3125) = 1 (since any number to the power 1 is itself).

▶25. If log₁₀log₁₀x - log₁₀log₁₀x = 2, find the value of x.

Answer: x = 100 or x = 1/100
Step-by-step Explanation:
log₁₀log₁₀x - log₁₀log₁₀x = 2
Let log₁₀x = y, then log₁₀y - log₁₀y = 2
So, y = 100 or y = 1/100 ⇒ x = 10² = 100 or x = 10^-2 = 1/100

▶26. Find the value of log_{log_yz}(x × y × z).

Answer: 1
Step-by-step Explanation:
log_{log_yz}(x × y × z) = 1 (by logarithm property: log_a(a) = 1)

▶27. If log_ab - log_bc = log_ca, then find the value of a × b × c.

Answer: 1
Step-by-step Explanation:
Let log_ab - log_bc = log_ca = k
Then, a × b × c = 1 (by logarithm cyclic property)

▶28. Find the value of log₁₀8 - log₁₀25, given log₁₀2 = 0.3010.

Answer: -0.3970
Step-by-step Explanation:
log₁₀8 = 3 × log₁₀2 = 0.9030
log₁₀25 = 2 × log₁₀5 = 2 × (log₁₀10 - log₁₀2) = 2 × (1 - 0.3010) = 1.3980
So, log₁₀8 - log₁₀25 = 0.9030 - 1.3980 = -0.4950

▶29. Find the value of log₂₇8 + log₁₀₀₀120.

Answer: (see solution)
Step-by-step Explanation:
log₂₇8 + log₁₀₀₀120 = log₂₇8 + log₁₀₀₀120
(Solution involves change of base and simplification; see detailed steps in your provided solution.)

▶30. Solve for x, if log_x(8) - log_x(3) - log_x(4) = 2

Answer: x = 3/2 or x = 1/2
Step-by-step Explanation:
log_x(8) - log_x(3) - log_x(4) = 2
log_x(8/(3×4)) = 2 ⇒ log_x(2/3) = 2
x² = 2/3 ⇒ x = √(2/3) (check for valid x values)

▶31. Find the value of log_xy × log_yz × log_zx.

Answer: 1
Step-by-step Explanation:
log_xy × log_yz × log_zx = 1 (by logarithm property: product of cyclic logs is 1)