AMC 10 · 2013 · #14
Grade 11 algebrapatternPick an answer.
The information arrives in the wrong shape: the condition is about logarithms, but the question asks about the number inside one. Tool #15 (Organize Information in More Ways) is primary — rewrite the arithmetic ladder of logarithms as a multiplicative ladder of the numbers themselves, then rewrite 162 and 1250 in prime-factored form, and the answer becomes visible with no decimals. Tool #4 (Introduce a Variable) names the step factor so the two known end numbers can be tied together in one equation. Tool #3 (Eliminate Possibilities) handles the one place this problem can bite: a fourth power hides a sign, so two real step factors solve the equation and one of them must be thrown out. Tool #7 (Identify Subproblems) splits the work honestly into two halves — first show what x is forced to be, then show a progression with that x actually exists.
Turn the log ladder into a number ladder
The ladder becomes a multiplying one.
Adding a fixed amount to logarithms is the same as multiplying by a fixed factor, so an evenly spaced list of logs sits on top of an evenly multiplied list of numbers.
Adding a fixed amount to logarithms is multiplying by a fixed factor, so evenly spaced logs sit on a geometric ladder.
▸ Why?
A logarithm counts how many times a base is used, so adding counts is multiplying values.
▸ Why?
An evenly spaced list climbs by the same fixed step every time, which is exactly the condition given.
Tie the two ends together
The two ends fix the ratio.
The two numbers you are given sit exactly four steps apart, so between them they already fix the fourth power of the step factor.
9.A-CED.A.1Introduce A VariableRead the fourth root off the primes
The primes make the fourth root obvious.
Prime factors expose a hidden fourth power that a decimal approximation would only blur.
8.EE.A.1Organize Information In More WaysKill the second fourth root
A positive base kills the negative root.
A fourth power erases the sign, so the sign has to be recovered from somewhere else — here from the fact that logarithms refuse negative inputs.
11.N-RN.A.2Eliminate PossibilitiesBuild the ladder and check it lands
Building the ladder gives 270, choice (B).
Writing out the actual five numbers turns an answer that was merely forced into one that is verified.
9.F-IF.A.3Identify SubproblemsWhen logarithms climb by equal steps, the numbers inside them grow by equal factors — so factor the two ends, 162 = 2 · 3⁴ and 1250 = 2 · 5⁴, and the step factor 5/3 falls straight out.
- Turn the log ladder into a number ladder
- Tie the two ends together
- Read the fourth root off the primes
- Kill the second fourth root
- Build the ladder and check it lands