2014 AMC 12B Problem 20

The logarithms are defined only when $x - 40 \gt 0$ and $60 - x \gt 0,$ so $40 \lt x \lt 60.$ Within this range the inequality becomes $(x-40)(60-x) \lt 100,$ which expands to $x^2 - 100x + 2500 \gt 0,$ i.e. $(x-50)^2 \gt 0.$ This holds for every $x \ne 50.$ The integers strictly between $40$ and $60$ except $50$ are $41, \ldots, 49$ and $51, \ldots, 59,$ which is $18$ values. Thus, the correct answer is B.