AMC 10 · 2022 · #14
Grade 11 algebraPick an answer.
The instinct is to reach for a calculator, and the instinct is wrong: each of these four logarithms is irrational, yet all five answer choices are exact rationals. So the problem is not asking for arithmetic — it is asking which combinations of these logarithms are clean even though the pieces are not. Tool #4 (Introduce a Variable) leads, and it leads twice. First it names a=log 5 and b=log 20, because those two never need to be known separately — only their sum matters, and their sum is exactly 2. Then it names t=log 2, because 5, 20, 8 and 0.25 are all 2s and 10s in disguise, so one letter converts the entire problem into ordinary algebra. Tool #7 (Identify Subproblems) does the sorting up front: the two cubes obey one set of rules and the lone product obeys another, so they get handled separately and are only added at the very end. Tool #5 (Look for a Pattern) supplies the identity that makes a³+b³ reachable from a+b alone. Tool #15 (Organize Information in More Ways) does the rewriting in the middle — 5=10/2, 20=2 · 10, 8=2³, 0.25=2⁻² — which is where the problem stops being about logarithms and starts being about a quadratic in one letter.
Split into two unlike pieces
Split into two unlike pieces.
Terms that obey different rules should be separated before anything is done to them, or one rule gets forced onto a term it was never meant for.
9.A-SSE.A.1Identify SubproblemsThe two logs sum to exactly 2
The two logarithms sum to exactly two.
Two logarithms that are individually ugly can have a beautiful sum, so always check what their arguments multiply to before doing anything else.
11.F-LE.A.4Introduce A VariableCube the sum, then correct it
Cube the sum, then correct it.
Anything symmetric in a and b can be rebuilt from just a+b and ab, so once the sum is known there is only one quantity left to chase.
Anything symmetric in two numbers can be rebuilt from just their sum and their product.
▸ Why?
Expanding a cubed sum spreads the multiplication out into the cubes plus a multiple of the product.
▸ Why?
The sum and the product are exactly what a quadratic's coefficients record about its two roots.
Rewrite everything using log 2
Rewrite everything with one logarithm.
When every number in sight is built from the same prime, one letter for that prime's logarithm turns a logarithm problem into ordinary algebra.
11.N-RN.A.1Organize Information In More WaysAdd the parts and watch t vanish
Adding makes the letter vanish, leaving 2.
If a messy quantity cancels itself out of the final line, that is usually the problem confirming you took the route it was built for.
9.A-SSE.A.2Identify SubproblemsBefore cubing anything, check what the numbers inside the logarithms multiply to: here 5 × 20=100 makes the two logarithms add to exactly 2, and the rest of the expression was built so the leftovers cancel.
- Split into two unlike pieces
- The two logs sum to exactly 2
- Cube the sum, then correct it
- Rewrite everything using log 2
- Add the parts and watch t vanish