Problem:
Square ABCD has area 36 , and AB is parallel to the x-axis. Vertices A,B, and C are on the graphs of y=loga​x,y=2loga​x, and y=3loga​x, respectively. What is a?
Answer Choices:
A. 63​
B. 3​
C. 36​
D. 6​
E. 6 Solution:
Let A=(p,loga​p) and B=(q,2loga​q). Then AB=6=∣p−q∣. Because AB is horizontal, loga​p=2loga​q=loga​q2, so p=q2. Thus ∣∣∣​q2−q∣∣∣​=6, and the only positive solution is q=3. Note that C=(q,3loga​q), so BC=6=loga​q, from which a6=q=3 and a=63​​.