AMC 10 · 2015 · #8
Grade 11 algebraPick an answer.
The only awkward object here is log₅ 2015, and the fastest way to defuse a logarithm is to name it: set x = log₅ 2015, which converts the whole log into the single plain equation 5^x = 2015 (Introduce a Variable). After that the obstacle is cosmetic — the power is written in base 625 while the useful fact is written in base 5 — so I rewrite 625 as 5⁴ to put everything over one base (Organize Information in More Ways). Once the bases agree, the outer 1/4 power exactly undoes the 4, and the stored equation 5^x = 2015 finishes it. Finally I bracket the value between two powers of 5 and check the five choices against that window (Eliminate Possibilities), which confirms the answer by a route that does not reuse the algebra.
Name the logarithm
The logarithm gets a name.
A logarithm is nothing but the exponent you were hunting for, so naming it turns scary notation into one ordinary equation.
11.F-LE.A.4Introduce A VariablePut everything over base 5
Everything rewrites over one base.
Two powers can only talk to each other once they speak the same base, and 625 is just 5 said four times.
9.A-SSE.A.2Organize Information In More WaysThe fourth root undoes the 4
The root undoes the power exactly.
Quadrupling an exponent and then taking a fourth root are exact opposites, so the exponent comes back untouched.
Quadrupling an exponent and then taking a fourth root are exact opposites, so the exponent comes back untouched.
▸ Why?
An exponent counts how many times the base is used, so raising a power to a power multiplies those counts.
▸ Why?
A root undoes a power exactly, so multiplying the count and then dividing it leaves it where it started.
Cash in the stored equation
The stored equation gives 2015.
Raising 5 to a base-5 logarithm simply hands the original number back — the two operations are built to cancel.
11.F-LE.A.4Introduce A VariableTrap the value between powers of 5
Bracketing the value confirms it, choice (D).
Pinning the exponent between 4 and 5 pins the value between two powers of 5, and only one choice fits in that gap.
11.N-RN.A.2Eliminate PossibilitiesWrite 625 as 5⁴ and the outer fourth root cancels that 4, leaving 5^log₅ 2015 — and a base-5 power sitting on a base-5 logarithm just hands the number back.
- Name the logarithm
- Put everything over base 5
- The fourth root undoes the 4
- Cash in the stored equation
- Trap the value between powers of 5