1985 AHSME Problem 24

A non-zero digit is chosen in such a way that the probability of choosing digit d is \log_{10}{(d+1)}-\log_{10}{d}. The probability that the digit 2 is chosen is exactly \frac{1}{2} the probability that the digit chosen is in the set

\mathrm{(A)\ } \{2, 3\} \qquad \mathrm{(B) \ }\{3, 4\} \qquad \mathrm{(C) \ } \{4, 5, 6, 7, 8\} \qquad \mathrm{(D) \ } \{5, 6, 7, 8, 9\} \qquad \mathrm{(E) \ }\{4, 5, 6, 7, 8, 9\}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.