AMC 10 · 2009 · #15

Grade 11 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Answer-Choice Plug-In

exponential-functionlogarithm-propertiesoptimizationinterval-arithmetic easier-related-problemextreme-principle ↑ Prerequisites: exponential-functionlogarithm-properties

📏 Long solution 💡 3 insights

Problem

Five equations share one shape and differ only in their base. Decide which one has the largest solution.

Pick an answer.

(A)

$3(1 + r)^x = 7$

(B)

$3(1 + r/10)^x = 7$

(C)

$3(1 + 2r)^x = 7$

(D)

$3(1 + \sqrt {r})^x = 7$

(E)

$3(1 + 1/r)^x = 7$

How to solve

Strategy Solve an Easier Related Problem

Solving five exponential equations and comparing five messy answers is the slow road. Tool #4 (Introduce a Variable) names the only thing that changes: write every option as 3b^x = 7 with base b. Tool #9 (Solve an Easier Related Problem) is primary, because that common form shows the largest x belongs to the smallest base, which replaces the question with a much easier one: which of r, r/10, 2r, √(r), 1/r is smallest on (0,3)? Tool #3 (Eliminate Possibilities) knocks the four rivals out one comparison at a time. Tool #14 (Extreme Principle) is what keeps the argument honest: only one of those comparisons can fail, so push r to the edge of its range and find the exact value where it would flip. That is where the hypothesis 0 ＜ r ＜ 3 earns its place, and it is why a single test value such as r=2 is not a proof.

1STEP 1

Reduce five equations to one form

All five reduce to one form where only the base differs.

3b^x=7 ⇔ b^x=7/3, b ∈ {1+r, 1+r/10, 1+2r, 1+√(r), 1+1/r}

2STEP 2

Bigger base means smaller x

A bigger base gives a smaller solution.

x=ln7/3/(ln b), 1 ＜ b₁ ＜ b₂ → x₁ ＞ x₂

3STEP 3

Rule out r, 2r, and root r

Three bases are ruled out immediately.

r/10 ＜ r ⇔ 1/10 ＜ 1; r/10 ＜ 2r ⇔ 1/10 ＜ 2; r/10 ＜ √(r) ⇔ r ＜ 100

4STEP 4

The one comparison that is tight

One comparison is genuinely tight and needs the given range.

r/10 ＜ 1/r ⇔ r² ＜ 10; 0 ＜ r ＜ 3 → r² ＜ 9 ＜ 10

5STEP 5

Push r to the edge

Pushing the range to its edge still holds.

r/10=1/r ⇔ r=√(10)≈ 3.162, 3 ＜ √(10)

6STEP 6

Smallest base, largest solution

The smallest base wins, choice (B).

x_max=log₁+r/107/3 → (B)

Answer

3(1 + r/10)^x = 7

Check with real numbers. At r=2 the bases are 3, 1.2, 5, 1+√(2)≈ 2.414, 1.5, and x=ln7/3/(ln b) gives about 0.77, 4.65, 0.53, 0.96, 2.09 — the r/10 option is far ahead. At r=0.1 it gives about 85.2 against 0.35 for the 1/r option. At r=2.9, close to the boundary, it is about 3.33 against 2.86 — still ahead, and the 1/r option is the runner-up, exactly as the boundary analysis predicts. The limiting behaviour also matches: as r→ 0^+ the base 1+r/10 tends to 1, so its solution blows up, while 1+1/r blows up and its solution shrinks toward 0. Finally, every solution came out positive, which it must, since 7/3 ＞ 1 forces x ＞ 0 for a base above 1. So (B) holds throughout the allowed range, not just at a sample point.

💡Key takeaway

All five equations climb from 3 up to 7; the one with the smallest base climbs slowest, so it needs the biggest exponent.

Reduce five equations to one form
Bigger base means smaller x
Rule out r, 2r, and root r
The one comparison that is tight
Push r to the edge
Smallest base, largest solution