AMC 10 · 2017 · #20

Grade 11 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Logic-Deduction / Elimination

logarithm-propertiespolynomial-rootszero-product-property convert-to-algebracasework ↑ Prerequisites: logarithm-propertiesexponents

📏 Medium solution 💡 3 insights

Problem

A logarithm raised to a high power must equal the logarithm of a high power. Count the pairs.

Pick an answer.

(A)

198

(B)

199

(C)

398

(D)

399

(E)

597

How to solve

Strategy Introduce a Variable

The exponent 2017 looks frightening, but the base b appears only inside the logarithms. Tool #4 (Introduce a Variable) is the spine: naming x = log_b a collapses the whole equation into x²⁰¹⁷ = 2017x, an equation with no a and no b left in it. Tool #13 (Convert to Algebra) then handles that equation as an ordinary polynomial, where factoring — not dividing — is what keeps every root. Tool #7 (Identify Subproblems) separates the two questions that are easy to blur together: how many values of x work, and how many pairs (a,b) each value of x produces. Tool #2 (Make a Systematic List) finishes by organising the pairs into three families, one per root, so nothing is double-counted.

1STEP 1

Name the logarithm

Naming the logarithm gives one plain equation.

x = log_b a → x²⁰¹⁷ = 2017x

2STEP 2

Factor, do not divide

Factoring keeps the zero solution.

x²⁰¹⁷ - 2017x = 0 → x(x²⁰¹⁶ - 2017) = 0 → x = 0 or x²⁰¹⁶ = 2017

3STEP 3

An even power gives two roots

The even power adds two more roots.

x²⁰¹⁶ = 2017 → x = ± 2017¹/2016, x ∈ {0, 2017¹/2016, -2017¹/2016}

4STEP 4

Turn each root into pairs

Each base turns each root into a pair.

log_b a = x ⇔ a = b^x, b ∈ {2,3,…,200} (200-2+1 = 199 bases)

5STEP 5

Count without double counting

Multiplying gives 597, choice (E).

3 × 199 = 597 → (E)

Answer

597

Spot-check one base. Take b = 2 and a = 1: the left side is (log₂ 1)²⁰¹⁷ = 0²⁰¹⁷ = 0 and the right side is log₂(1²⁰¹⁷) = log₂ 1 = 0, so it works. Take a = 2^c with c = 2017¹/2016: the left side is c²⁰¹⁷ = c · c²⁰¹⁶ = 2017c and the right side is 2017log₂ a = 2017c, so it works too, and the negative root works by the same computation because 2017 is odd. The wrong answers are exactly the near-misses: 398 = 2 × 199 is what you get by dividing by x and losing the root x = 0, and 199 is what you get by keeping only x = 0. Keeping all three roots gives 597, which matches choice (E).

💡Key takeaway

Give the logarithm a name so the giant exponent shrinks to one equation, factor instead of dividing so the zero root survives, then multiply the number of roots by the number of bases.

Name the logarithm
Factor, do not divide
An even power gives two roots
Turn each root into pairs
Count without double counting

Name the logarithm

Factor, do not divide

An even power gives two roots

Turn each root into pairs

Count without double counting

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