AMC 10 · 2018 · #14
Grade 11 algebraPick an answer.
The equation is awkward because x hides inside two different bases, and there is no rule for comparing a base-3x logarithm with a base-2x logarithm directly. The fix is to name the quantity the two sides share. Call the common value y. Each logarithm statement then becomes an ordinary exponential statement, (3x)^y = 4 and (2x)^y = 8, with x back in a normal position. From there the two right-hand sides can be pushed to a common number, the exponent y cancels, and what is left is a plain polynomial equation in x.
Name the shared value and drop the logs
Naming the shared value strips the logarithms.
A logarithm is just an exponent wearing a disguise, so naming that exponent turns both statements into ordinary powers.
11.F-LE.A.4Introduce A VariablePush both right sides to the same number
Push both right sides to the same number.
Two equations only talk to each other once their right sides are the same number, and 64 is the meeting point of 4 and 8.
11.N-RN.A.1Organize Information In More WaysCancel the exponent y
Then the exponent cancels.
Raising to a fixed nonzero power never sends two different positive numbers to the same place, so equal powers force equal bases.
Raising to a fixed nonzero power never sends two different positive numbers to the same place, so equal powers force equal bases.
▸ Why?
Two equal powers of the same base must have equal exponents, and the same reading works the other way.
▸ Why?
An exponent counts how many times a factor is used, so a bigger base gives a strictly bigger power.
Solve the polynomial equation
Solve the leftover polynomial.
Dividing by x² looks tempting, but factoring instead shows exactly which root the domain throws away.
11.A-APR.B.3Convert To AlgebraCheck the domain and add p and q
Checking the domain and adding gives 31.
A candidate root is only a solution once it survives the original equation's domain restrictions.
6.NS.B.4Eliminate PossibilitiesWhen an unknown is stuck inside a logarithm base, name the exponent the two sides share — the logarithms turn into ordinary powers, and the shared exponent cancels itself out.
- Name the shared value and drop the logs
- Push both right sides to the same number
- Cancel the exponent y
- Solve the polynomial equation
- Check the domain and add p and q