Can we solve these three simultaneous log equations?

Can we solve these three simultaneous log equations?

Question

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.

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1. is satisfied by two values of $a$.

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1. is satisfied by infinitely many values of $a$.

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1. is contradictory.

Suggestion

Could we rewrite the equations using powers rather than logarithms?

Solution

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).