Three positive numbers \(a\), \(b\), \(c\) satisfy \[\log_b a = 2, \qquad \log_b (c-3) = 3, \qquad \log_a (c+5) = 2.\] This information

  1. specifies \(a\) uniquely.

  2. is satisfied by two values of \(a\).

  3. is satisfied by infinitely many values of \(a\).

  4. is contradictory.

If we write the three equations in exponential form, we get \[a=b^2, \qquad c-3 = b^3, \qquad c+5 = a^2.\] We can then eliminate \(a\) and \(c\) by substituting the first and second equations into the third, to get \[b^3 + 3 + 5 = b^4,\] which rearranges to \[b^3(b-1)=8.\]

We require \(b\) to be positive, so we are only interested in positive solutions of this equation.

We note that \(b^3(b-1)\) is negative for \(0<b<1\), and then for \(b>1\), both \(b^3\) and \(b-1\) are positive and increasing (without bound).

So there is only one positive solution to this equation (in fact by looking at it we can easily spot that \(b=2\)), so there is only one possible value for \(a\) and for \(c\) (namely \(a=4\) and \(c=11\)).

The answer is (a).