AMC 10 · 2010 · #11
Grade 11 algebraPick an answer.
Tool #11 (Work Backwards): the requested form is the real instruction. Since a logarithm is an exponent, x = log_b 7⁷ says the same thing as b^x = 7⁷, so I know the exact shape the equation has to be pushed into — one power with exponent x on one side, 7⁷ alone on the other — and the base I end up with is the answer. Tool #15 (Organize Information in More Ways): the given equation has x in two places and 7 in two roles, so I regroup the powers — split 7^x+7, then divide by 7^x — until every x sits inside a single power. Tool #3 (Eliminate Possibilities): the problem presents five candidate bases, so I check that the base is forced, that is, that no second base could give the same x; that argument disposes of the other four choices at once instead of one at a time.
Turn the target form into an equation
The required form is just an exponential statement.
A logarithm is just an exponent wearing a different name, so the answer form is a hidden exponential equation.
11.F-LE.A.4Work BackwardsSplit the constant off the left side
The added seven splits off as a constant factor.
Adding in the exponent is multiplying in the value, so a sum upstairs breaks apart into a product.
Adding in the exponent is multiplying in the value, so a sum upstairs breaks apart into a product.
▸ Why?
An exponent counts how many times a factor is used, so combining counts is combining factors.
▸ Why?
Once the constant piece is separated out, it multiplies everything that is left as a single factor.
Gather every x into one power
Dividing gathers every unknown into one power.
Two powers sharing an exponent can be merged into one power by dividing the bases instead.
9.A-SSE.A.2Organize Information In More WaysCheck that exactly one x exists
That power is increasing, so the solution is unique.
An exponential with base above 1 climbs without ever repeating a value, so it can hit 7⁷ only once.
9.F-IF.A.1Eliminate PossibilitiesShow the base is forced, then name it
Matching the two forms forces the base 8/7.
With the inside of the logarithm nailed down at 7⁷, the base and the output move together one-to-one, so no second base is available.
11.F-IF.C.8Eliminate PossibilitiesA logarithm is an exponent in disguise, so read x = log_b 7⁷ as b^x = 7⁷, reshape the equation until it matches that pattern, and the base is sitting in plain sight.
- Turn the target form into an equation
- Split the constant off the left side
- Gather every x into one power
- Check that exactly one x exists
- Show the base is forced, then name it