n = 24: log₂(24) ≈ 4.58, 0.2×24 = 4.8 → 4.58 < 4.8 → B faster - Decision Point
Understanding the Inequality: Why log₂(24) ≈ 4.58 is Less Than 0.2×24 = 4.8
Understanding the Inequality: Why log₂(24) ≈ 4.58 is Less Than 0.2×24 = 4.8
When dealing with mathematical inequalities, precision and clarity help reveal important truths. Consider the expression involving log base 2:
log₂(24) ≈ 4.58, while 0.2 × 24 = 4.8, leading to the clean comparison:
Understanding the Context
> 4.58 < 4.8
This seemingly simple comparison highlights a key insight: even though one value is derived logarithmically and the other is a simple decimal multiplication, the inequality log₂(24) < 0.2×24 holds true.
Why Is log₂(24) Approximately 4.58?
The value log₂(24) represents the exponent needed to raise base 2 to equal 24. Since:
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Key Insights
- 2⁴ = 16
- 2⁵ = 32, and
- 24 lies between 16 and 32,
the logarithm must be between 4 and 5. Precise calculation gives:
log₂(24) ≈ 4.58496, rounding to 4.58 for simplicity.
This means log₂(24) reflects how many base-2 doublings are needed to reach 24.
Why Is 0.2×24 Exactly 4.8?
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Multiplication by 0.2 is equivalent to multiplying by 1/5. So:
0.2 × 24 = 24 ÷ 5 = 4.8
This straightforward computation avoids approximation and provides clarity on the right-hand side of the inequality.
Comparative Insight: 4.58 < 4.8
Because:
- log₂(24) ≈ 4.58
- 0.2×24 = 4.8,
we see definitively that:
log₂(24) < 0.2×24
This difference shows how logarithmic growth rates differ from linear scaling—logarithmic functions grow more slowly than linear functions for values above 1.
