2007 AMC 12B Problem 17

Problem 17 of 25IntermediateAlgebra

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,\frac{1}{b}\}?

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Solution

For

b\gt0,

we have

b\lt10^b,

hence

\log_{10}b\lt b.

b=1,

then

a=0.

b\gt1,

then

0\lt\dfrac{\log_{10}b}{b^2}\lt1,

a

could not be an integer. Hence

0\lt b\lt1,

\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.