Suppose that 4^x_1 = 5, 5^x_2 = 6, 6^x_3 = 7, …, 127^x_124 = 128. What is x_1 x_2 \cdots x_124?

2005 AMC 12B Problem 13 — Solution

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13.

Suppose that $4^{x_1} = 5,$ $5^{x_2} = 6,$ $6^{x_3} = 7, \ldots, 127^{x_{124}} = 128.$ What is $x_1 x_2 \cdots x_{124}?$

$2$

$\dfrac{5}{2}$

$3$

$\dfrac{7}{2}$

$4$

From $4^{x_1} = 5$ we get $x_1 = \log_4 5,$ and in general $x_k = \log_{k+3}(k+4).$

The product telescopes: $\begin{aligned} &x_1 x_2 \cdots x_{124} \\ &= \log_4 5 \cdot \log_5 6 \cdots \log_{127} 128 \\ &= \log_4 128. \end{aligned}$

Since $128 = 2^7$ and $4 = 2^2,$ this equals $\dfrac{7\log 2}{2\log 2} = \dfrac72.$

Thus, the correct answer is D.