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 $\{0,1,a,b,1/b\}$?
$\mathrm{(A)}\ 0 \qquad \mathrm{(B)}\ 1 \qquad \mathrm{(C)}\ a \qquad \mathrm{(D)}\ b \qquad \mathrm{(E)}\ \frac{1}{b}$
Note that if $a$ is positive, then, the equation will have no solutions for $b$. This becomes more obvious by noting that at $b=1$, $ab^2 > \log_{10} b$. The LHS quadratic function will increase faster than the RHS logarithmic function, so they will never intersect.
This puts $a$ as the smallest in the set since it must be negative.
Checking the new equation: $-b^2 = \log_{10}b$
Near $b=0$, $-b^2 > \log_{10} b$ but at $b=1$, $-b^2  1$
This makes our set (ordered)
$\{a,0,b,1,1/b\}$
The median is $b \Rightarrow \mathrm {(D)}$