Logarithm practice worksheet with equations and spaces for solutions.

Here is the complete solution to all the logarithmic expressions on the worksheet, with explanations for each.

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Key Logarithmic Rules Used:

1. Definition: `log_b(a) = c` means `b^c = a`.
2. log_b(1) = 0 for any base `b > 0`, `b ≠ 1` (since any number to the power 0 is 1).
3. log_b(b) = 1 (since `b^1 = b`).
4. log_b(b^k) = k.
5. log_b(1/b^k) = -k (since `b^(-k) = 1/b^k`).

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## ✔ SOLUTIONS:

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Left Column:

1. log₇(49)
→ 49 = 7² → log₇(7²) = 2

2. log₂(32)
→ 32 = 2⁵ → log₂(2⁵) = 5

3. log₁₂(1)
→ Any log of 1 is 0 → 0

4. log₄(1/16)
→ 1/16 = 4⁻² → log₄(4⁻²) = -2

5. log₃(3)
→ log_b(b) = 1 → 1

6. log₄(16)
→ 16 = 4² → log₄(4²) = 2

7. log₂(1/2)
→ 1/2 = 2⁻¹ → log₂(2⁻¹) = -1

8. log₁₀(100)
→ 100 = 10² → log₁₀(10²) = 2

9. log₄(4)
→ log_b(b) = 1 → 1

10. log₂(2)
→ log_b(b) = 1 → 1

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Right Column:

1. log₄(1)
→ log_b(1) = 0 → 0

2. log₃(81)
→ 81 = 3⁴ → log₃(3⁴) = 4

3. log₁₀(1/1000)
→ 1/1000 = 10⁻³ → log₁₀(10⁻³) = -3

4. log₇(7)
→ log_b(b) = 1 → 1

5. log₃(9)
→ 9 = 3² → log₃(3²) = 2

6. log₇(1)
→ log_b(1) = 0 → 0

7. log₁₀(1/100)
→ 1/100 = 10⁻² → log₁₀(10⁻²) = -2

8. log₇(1/7)
→ 1/7 = 7⁻¹ → log₇(7⁻¹) = -1

9. log₈(1)
→ log_b(1) = 0 → 0

10. log₇(1/49)
→ 1/49 = 7⁻² → log₇(7⁻²) = -2

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## 📝 Final Answer Sheet:

```
log₇(49) = 2 log₄(1) = 0
log₂(32) = 5 log₃(81) = 4
log₁₂(1) = 0 log₁₀(1/1000) = -3
log₄(1/16) = -2 log₇(7) = 1
log₃(3) = 1 log₃(9) = 2
log₄(16) = 2 log₇(1) = 0
log₂(1/2) = -1 log₁₀(1/100) = -2
log₁₀(100) = 2 log₇(1/7) = -1
log₄(4) = 1 log₈(1) = 0
log₂(2) = 1 log₇(1/49) = -2
```

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✔ All answers are exact integers — no calculators needed! Just apply the basic definition and properties of logarithms.

Let me know if you’d like a printable version or step-by-step breakdowns for any specific problem!