2005 AMC 12B Problems/Problem 13

Problem

Suppose that $4^{x_1}=5$ , $5^{x_2}=6$ , $6^{x_3}=7$ , ... , $127^{x_{124}}=128$ . What is $x_1x_2...x_{124}$ ?

$\mathrm{(A)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}$

Solution

We see that we can re-write $4^{x_1}=5$ , $5^{x_2}=6$ , $6^{x_3}=7$ , ... , $127^{x_{124}}=128$ as $\left(...\left(\left(\left(4^{x_1}\right)^{x_2}\right)^{x_3}\right)...\right)^{x_{124}}=128$ by using substitution. By using the properties of exponents, we know that $4^{x_1x_2...x_{124}}=128$ .

$\begin{align*}4^{x_1x_2...x_{124}}=128\\2^{2x_1x_2...x_{124}}=2^7\\2x_1x_2...x_{124}=7\\x_1x_2...x_{124}=\dfrac{7}{2}\end{align*}$ (Error compiling LaTeX. Unknown error_msg)

Therefore, the answer is $\boxed{\mathrm{(D)}\,\dfrac{7}{2}}$

2005 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 12	Followed by Problem 14
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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2005 AMC 12B Problems/Problem 13

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