2005 AMC 10B Problems/Problem 17

Problem

Suppose that $4^a = 5$ , $5^b = 6$ , $6^c = 7$ , and $7^d = 8$ . What is $a * b * c * d$ ?

$\mathrm{(A)} 1 \qquad \mathrm{(B)} \frac{3}{2} \qquad \mathrm{(C)} 2 \qquad \mathrm{(D)} \frac{5}{2} \qquad \mathrm{(E)} 3$

Solution

$8=7^d$ $8=\left(6^c\right)^d$ $8=\left(\left(5^b\right)^c\right)^d$ $8=\left(\left(\left(4^a\right)^b\right)^c\right)^d$ $8=4^{a\cdot b\cdot c\cdot d}$ $2^3=2^{2\cdot a\cdot b\cdot c\cdot d}$ $3=2\cdot a\cdot b\cdot c\cdot d$ $a\cdot b\cdot c\cdot d=\boxed{\mathrm{(B)}\ \dfrac{3}{2}}$

2005 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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2005 AMC 10B Problems/Problem 17

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