Problem:
Let be the sum of all positive real numbers for which
Which of the following statements is true?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Answer (D): Taking the logarithm to base of each side of the equation gives
This is equivalent to . The shapes of the graphs of the two functions in this latter equation reveal where they intersect. The logarithmic function is increasing and concave down, while the exponential function is increasing and concave up. Hence their graphs can intersect in at most two points. One straightforward solution is . To see that there is in fact another solution, observe that , whereas . Therefore a solution must exist between 2 and 4 . It follows that the sum of the solutions must be in the interval . Thus . See the graphs below.
The problems on this page are the property of the MAA's American Mathematics Competitions