Problem:
What is the product of all real numbers x such that the distance on the number line between log6x and log69 is twice the distance on the number line between log610 and 1 ?
Answer Choices:
A. 10
B. 18
C. 25
D. 36
E. 81
Solution:
The distance between log6x and log69 is ∣log6x−log69∣=∣∣∣log6(9x)∣∣∣, which is either log6(9x) or log6(x9). The distance between log610 and 1 is ∣log610−log66∣=∣∣∣log6(610)∣∣∣, which is log6(35) because distance is nonnegative. Because twice log6(35) is log6(35)2, it follows that
9x=925 or x9=925
Solving these equations gives, respectively, x=25 and x=2581. Thus the requested product is 25⋅2581=(E)81.
The problems on this page are the property of the MAA's American Mathematics Competitions