2007 AMC 12B Problem 17

Below is the professionally curated solution for Problem 17 of the 2007 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12B solutions, or check the answer key.

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Concepts:logarithm inequality median (data)

Difficulty rating: 2060

17.

If $a$ is a nonzero integer and $b$ is a positive number such that $ab^2=\log_{10}b,$ what is the median of the set $\{0,1,a,b,1/b\}?$

$0$

$1$

$a$

$b$

$\dfrac{1}{b}$

Solution:

Because $b\lt10^b$ for all $b\gt0,$ it follows that $\log_{10}b\lt b.$ If $b\ge1,$ then $0\lt\dfrac{\log_{10}b}{b^2}\lt1,$ so $a$ could not be a nonzero integer.

Hence $0\lt b\lt1,$ so $\log_{10}b\lt0$ and $a=\dfrac{\log_{10}b}{b^2}\lt0.$ Thus $a\lt0\lt b\lt1\lt\dfrac1b,$ and the middle value of the sorted set is $b.$

Thus, the correct answer is D.

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