AMC 10 · 2022 · #11

Grade 11 algebra
Archetype Pair-Sum Invariant └ Sub-archetype Uniform Pair-Sum Recovery
logarithm-propertiesabsolute-valuecaseworkexponents convert-to-algebracaseworkidentify-subproblems ↑ Prerequisites: logarithm-properties
📏 Medium solution 💡 2 insights
Problem
Every quantity here is a point on one number line. One logarithm must sit exactly twice as far from a second point as a third point sits from a fourth. Find every real number that does this, then multiply those values together.

Pick an answer.

(A)
10
(B)
18
(C)
25
(D)
36
(E)
81
How to solve
Strategy Convert to Algebra

The sentence is geometric — points and distances — but it lives on a number line, and distance on a number line is the absolute value of a difference. Tool #13 (Convert to Algebra) turns the whole sentence into one equation with absolute-value bars. Then tool #7 (Identify Subproblems) splits the work: the right-hand side is a fixed number, so measure it first, alone, using log rules. Once that number is known, tool #2 (Make a Systematic List) handles the absolute value the honest way — one solution above log₆ 9, one below, and neither may be dropped. Finally the question asks for a product, not for each root, so tool #16 (Change Focus) points at the sum of the two logarithms instead of the roots themselves; that sum is where the answer falls out immediately.

1STEP 1

Rewrite each distance as absolute value

Write both distances as absolute values.

|log₆ x - log₆ 9| = 2|log₆ 10 - 1|, x ＞ 0
2STEP 2

Measure the right-hand distance first

Reduce the right-hand distance to one logarithm.

2|log₆ 10 - log₆ 6| = 2log₆ 10/6 = 2log₆ 5/3 = log₆(5/3)² = log₆ 25/9
3STEP 3

List both sides of the center

List both sides of the centre.

log₆ x = log₆ 9 ± log₆ 25/9 ⟹ log₆ x = log₆ 25 or log₆ x = log₆ 81/25 ⟹ x = 25 or x = 81/25
4STEP 4

Multiply, and see why the fractions cancel

Multiplying cancels the fractions, giving 81.

25 · 81/25 = 81; log₆ x₁ + log₆ x₂ = (log₆ 9 + d) + (log₆ 9 - d) = 2log₆ 9 = log₆ 81 ⟹ x₁ x₂ = 81 → (E)
Answer
81
Check the two roots numerically. log₆ 10 ≈ 1.2851, so the right-hand distance is 2(1.2851 - 1) = 0.5702. Also log₆ 9 ≈ 1.2263. For x = 25: log₆ 25 ≈ 1.7965 and 1.7965 - 1.2263 = 0.5702, a match. For x = 81/25 = 3.24: log₆ 3.24 ≈ 0.6561 and 1.2263 - 0.6561 = 0.5702, also a match. Both roots are positive, so both logs exist, and 25 · 3.24 = 81 confirms choice (E). The wrong choices are all reachable by a specific slip: (C) 25 is the answer if only the farther root is found and the nearer one 81/25 is forgotten; (D) 36 = 6² squares the base instead of the center 9; (B) 18 = 2 · 9 doubles the center directly instead of doubling a logarithm; (A) 10 just echoes the 10 inside the problem. Only 81 = 9² agrees with the symmetry of the two roots about log₆ 9.
💡Key takeaway

A distance between logarithms is itself a logarithm, and two points sitting equally far above and below log₆ 9 always multiply back to 9².

  • Rewrite each distance as absolute value
  • Measure the right-hand distance first
  • List both sides of the center
  • Multiply, and see why the fractions cancel

More like this

Same archetype — closest grade level first.