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smc

algebra senior

Problem

If

y = (lo g_{2} 3) (lo g_{3} 4) \dots (lo g_{n} [n + 1]) \dots (lo g_{31} 32)

then

(A)

4 < y < 5

(B)

y = 5

(C)

5 < y < 6

(D)

y = 6

Solution

Note that

lo g_{a} b = \frac{lo g b}{lo g a}

. Then

y = (\frac{lo g 3}{lo g 2}) (\frac{lo g 4}{lo g 3}) \dots (\frac{lo g 32}{lo g 31}) = \frac{lo g 32}{lo g 2} = lo g_{2} 32 = y = 5

.

Final answer

B