AMC 10 · 2021 · #9

Grade 11 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
logarithm-propertiesexponentsprime-factorizationformula-substitution convert-to-algebraidentify-subproblems ↑ Prerequisites: logarithm-propertiesexponents
📏 Medium solution 💡 3 insights
Problem
Evaluate the difference of two fractions. The two numerators are base-two logarithms while the two denominators are logarithms in base forty and base twenty, so the bases have to be reconciled before anything can be combined.

Pick an answer.

(A)
0
(B)
1
(C)
$\frac54$
(D)
2
(E)
$\log_2 5$
How to solve
Strategy Introduce a Variable

Nothing here can be evaluated exactly by hand: log₂ 5 is an ugly irrational number. But it is the only ugly piece in the whole expression, because 80, 40, 160, and 20 are all a power of 2 times 5. So I give that one ugly piece a name, a = log₂ 5, and every logarithm turns into something like 4 + a. First, though, I rewrite the two denominators so that every logarithm has base 2 (Organize Information in More Ways) — dividing by log₄₀ 2 is the same as multiplying by log₂ 40. After that the whole thing is ordinary algebra in a.

1STEP 1

Flip the base-40 and base-20 logs

Flip the logarithms with the other bases.

log₄₀ 2 = 1/(log₂ 40) → (log₂ 80)/(log₄₀ 2) = log₂ 80 · log₂ 40, (log₂ 160)/(log₂₀ 2) = log₂ 160 · log₂ 20
2STEP 2

Name the only ugly piece

Give the messy piece a name.

a = log₂ 5 → log₂ 80 = 4 + a, log₂ 40 = 3 + a, log₂ 160 = 5 + a, log₂ 20 = 2 + a
3STEP 3

Expand both products

Expand both products.

(4 + a)(3 + a) = a² + 7a + 12, (5 + a)(2 + a) = a² + 7a + 10
4STEP 4

Subtract and watch a vanish

Subtracting makes the letter vanish, leaving 2.

(a² + 7a + 12) - (a² + 7a + 10) = 12 - 10 = 2
Answer
2
A numerical check confirms it. Using log₂ 5 ≈ 2.3219: log₂ 80 ≈ 6.3219, log₂ 40 ≈ 5.3219, log₂ 160 ≈ 7.3219, log₂ 20 ≈ 4.3219. Then 6.3219 × 5.3219 ≈ 33.645 and 7.3219 × 4.3219 ≈ 31.645, and the difference is 2.000. The answer being a clean integer is also believable in structure: 80 · 40 = 3200 = 160 · 20, so the two halves are built from number pairs with the same product, which is exactly why the a² and 7a terms match and cancel. Choice (E), log₂ 5, is the trap for anyone who cancels carelessly and leaves one copy of a behind; the expansion shows both copies go. This matches choice (D).
💡Key takeaway

When one ugly number shows up everywhere, give it a letter instead of computing it — here log₂ 5 cancels itself out and the answer is just 2.

  • Flip the base-40 and base-20 logs
  • Name the only ugly piece
  • Expand both products
  • Subtract and watch a vanish

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