Can we solve these three simultaneous log equations?

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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.

Could we rewrite the equations using powers rather than logarithms?