Example exercises from Practice 7-3, including logarithmic equations, exponents, and simplifications.

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Practice 7-3: Example Exercises

These are logarithmic and exponential expressions that test your understanding of:
- Logarithm definitions
- Properties of exponents
- Evaluating logs and powers

Let’s go through each one:

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#### 1. log₂8 = 3
✔ True
Because:
2³ = 8 → So log₂8 = 3

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#### 2. log₃81 = 4
✔ True
Because:
3⁴ = 81 → So log₃81 = 4

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#### 3. log₄64 = 3
✔ True
Because:
4³ = 64 → So log₄64 = 3

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#### 4. log₅625 = 4
✔ True
Because:
5⁴ = 625 → So log₅625 = 4

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#### 5. log(10,000) = 4
✔ True (assuming base 10)
Because:
10⁴ = 10,000 → So log₁₀(10,000) = 4

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#### 6. log₂(1/8) = -3
✔ True
Because:
2⁻³ = 1/8 → So log₂(1/8) = -3

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#### 7. log₂₇9 = 2/3
✔ True
We rewrite both sides with same base:
27 = 3³, 9 = 3²
So:
log₂₇9 = log₍₃³₎(3²)
Use change of base:
= (log₃(3²)) / (log₃(3³)) = (2) / (3) = 2/3

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#### 8. log₁₆8 = 3/4
✔ True
16 = 2⁴, 8 = 2³
So:
log₁₆8 = log₍₂⁴₎(2³) = (log₂(2³)) / (log₂(2⁴)) = 3/4

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#### 9. 2⁴ = 16
✔ True
2×2×2×2 = 16

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#### 10. 3⁻³ = 1/27
✔ True
3⁻³ = 1/(3³) = 1/27

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#### 11. 10² = 100
✔ True
10×10 = 100

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#### 12. 5³ = 125
✔ True
5×5×5 = 125

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#### 13. 8² = 64
✔ True
8×8 = 64

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#### 14. 9¹ = 9
✔ True
Any number to the power 1 is itself.

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#### 15. 12⁰ = 1
✔ True
Any non-zero number raised to 0 is 1.

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#### 16. 23⁰ = 1
✔ True
Same as above.

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#### 17. 5⁻¹ = 1/5
✔ True
Negative exponent means reciprocal: 5⁻¹ = 1/5

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#### 18. 7¹ = 7
✔ True

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#### 19. 2⁶ = 64
✔ True
2⁶ = 64

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#### 20. 6² = 36
✔ True
6×6 = 36

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#### 21. 2
This seems incomplete — likely a typo or missing context. Possibly meant to be part of a larger expression? Let’s assume it's just stating the number 2.

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#### 22. 1
Likely a value for a previous expression?

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#### 23. 0
Again, possibly a result from a logarithm or exponent problem.

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#### 24. 2
Another standalone number.

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#### 25. 1
Possibly a result.

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#### 26. 0
Could be from something like log₁(1), or 0^x (if x > 0).

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#### 27. 1/2
Maybe from log₄(2) = 1/2 because 4^(1/2) = 2.

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#### 28. 1/4
Could be from 2⁻² or 4⁻¹.

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#### 29. -2
Possibly from log₂(1/4) = -2 since 2⁻² = 1/4.

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#### 30. -3
From log₂(1/8) = -3

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#### 31. 2
Could be log₄(16) = 2 since 4² = 16

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#### 32. 5
Maybe log₅(3125) = 5? Since 5⁵ = 3125

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#### 33. 2.8
Possibly a decimal approximation of a logarithm, e.g., log₁₀(630) ≈ 2.8

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#### 34. 5.5
Maybe log₁₀(316,227) ≈ 5.5? Or from scientific notation?

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#### 35. 4.9
Likely an approximate log value, e.g., log₁₀(79,432) ≈ 4.9

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#### 36. 4.2
Similarly, could be log₁₀(15,848) ≈ 4.2

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#### 37. 4.0 × 10⁻²
That’s 0.04 — a small decimal number

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#### 38. 1.0 × 10⁻³
That’s 0.001

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✔ Summary

The first 20 items are true statements involving logarithms and exponents. The last 18 items (21–38) appear to be numerical answers to other problems (possibly fill-in-the-blank or multiple-choice), but without the original questions, we can only interpret them as possible results.

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🔍 Common Patterns

| Pattern | Example |
|--------|--------|
| log_b(b^n) = n | log₂(8) = 3 because 2³ = 8 |
| b⁰ = 1 | 12⁰ = 1 |
| b⁻ⁿ = 1/bⁿ | 3⁻³ = 1/27 |
| Change of Base: logₐb = log_c b / log_c a | log₂₇9 = log₃9 / log₃27 = 2/3 |

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🧠 Tips for Solving These Problems

1. Know your powers: Memorize squares, cubes, powers of 2, 3, 5, 10.
2. Use the definition: log_b(a) = c ⇔ bᶜ = a
3. Apply exponent rules: a⁻ⁿ = 1/aⁿ, a⁰ = 1, (aᵐ)ⁿ = a^(mn)
4. Use change of base when needed: log_b(a) = log(a)/log(b)

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