AMC 10 · 2024 · #8

Grade 11 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
logarithm-propertiesfraction-arithmeticexponents convert-to-algebrawork-backwards ↑ Prerequisites: logarithm-propertiesfraction-arithmetic
📏 Short solution 💡 3 insights
Problem
One number x makes the following true: take log₂ x and log₃ x, multiply them, divide that product by their sum, and the result is exactly 2. Find that x.

Pick an answer.

(A)
25
(B)
32
(C)
36
(D)
42
(E)
48
How to solve
Strategy Organize Information in More Ways

Attacking this equation head-on means cross-multiplying into log₂ x · log₃ x = 2(log₂ x + log₃ x), which looks like a two-variable problem and stalls. Tool #15 (Organize Information in More Ways) says: if the arrangement is fighting you, rewrite it. Here the useful rearrangement is turning the equation upside down, because a sum sitting on top of a product splits into two clean reciprocals. Tool #16 (Change Focus) then handles the payoff: a reciprocal of a logarithm is a logarithm with the base and the input traded, so instead of asking about base 2 and base 3 we ask about base x — and in base x the two terms merge into one. Tool #11 (Work Backwards) finishes: once a single logarithm is pinned to a value, run the definition in reverse to recover x.

1STEP 1

Flip both sides upside down

Flipping makes the right side one half.

(log₂ x + log₃ x)/(log₂ x · log₃ x) = 1/2
2STEP 2

Split the fraction into two

Each term becomes the reciprocal of a log.

(log₂ x)/(log₂ x · log₃ x) + (log₃ x)/(log₂ x · log₃ x) = 1/(log₃ x) + 1/(log₂ x) = 1/2
3STEP 3

Swap the base of each log

A reciprocal is the log with bases swapped.

1/(log_b x) = log_x b ⟹ log_x 3 + log_x 2 = 1/2
4STEP 4

Add the logs into one

They merge into log base x of six.

log_x 2 + log_x 3 = log_x (2 · 3) = log_x 6 = 1/2
5STEP 5

Undo the log to find x

Its square root is six, so x is 36.

log_x 6 = 1/2 ⟹ x¹/2 = 6 ⟹ x = 6² = 36 → (C)
Answer
36
Substitute x = 36 back into the original expression. log₂ 36 = (ln 36)/(ln 2) ≈ 3.5835/0.6931 ≈ 5.1699 and log₃ 36 = 3.5835/1.0986 ≈ 3.2619. Their product is about 16.863 and their sum is about 8.432, and 16.863/8.432 ≈ 2.000, so the equation holds. The structure also confirms it: the answer came out as (2 · 3)², the product of the two bases squared, and 2 was the target value — the general fact is that (log_a x · log_b x)/(log_a x + log_b x) = k forces x = (ab)^k. A quick check that no other choice works: for x = 32, log₂ 32 = 5 and log₃ 32 ≈ 3.1546, giving 15.773/8.1546 ≈ 1.934, not 2.
💡Key takeaway

When logs are stuck in an ugly fraction, flip the whole equation over: the reciprocal of log₂ x is just log_x 2, and once both logs share base x they merge into log_x 6.

  • Flip both sides upside down
  • Split the fraction into two
  • Swap the base of each log
  • Add the logs into one
  • Undo the log to find x

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