What is the value of _37_59_711_913_2125_2327?

From the Change of Base Formula, we have _i=3^13 (2i+1)_i=1^11 (2i+1) = 25 27 3 5 = (2 5)(3 3) 3 5 = 6.

smc

algebra senior

Problem

What is the value of

lo g_{3} 7 \cdot lo g_{5} 9 \cdot lo g_{7} 11 \cdot lo g_{9} 13 \dots lo g_{21} 25 \cdot lo g_{23} 27 ?

(A)

3

(B)

3 lo g_{7} 23

(C)

6

(D)

9

Solution

From the Change of Base Formula, we have

\frac{\prod _{i = 3}^{13} lo g ( 2 i + 1 )}{\prod _{i = 1}^{11} lo g ( 2 i + 1 )} = \frac{lo g 25 \cdot lo g 27}{lo g 3 \cdot lo g 5} = \frac{( 2 lo g 5 ) \cdot ( 3 lo g 3 )}{lo g 3 \cdot lo g 5} = 6 .

Final answer