Consider a geometric progression and an arithmetic progression with the following characteristics:
1- The geometric progression has a term equal to $1$ and the ratio is positive real number, $q,$ different than $1.$
2- The arithmetic progression has a term equal $0$ and the ratio, $r$, is a rational number which does not equal zero.
3- The two sequences are in bijective correspondence, such that the term $1$ of the geometric progression is associated with the term $0$ from the arithmetic progression.
Consider that the number $A$ in the arithmetic progression is associated with the number $23$ of the geometric progression. Consider also a number $B$ of the arithmetic progression associated with the number $30$ in the geometric progression.
What is the value of the Logarithm of $A$ times $B$?
So far I have written the terms of the GP and AP as such: $$GP: 1, q, q^2, q^3, \ldots, 23, \ldots, 30, \ldots$$ $$GA: 0, r, 2r, 3r, \ldots, A, \ldots, B, \ldots$$
Where: \begin{align} q^m &= 23\\ q^n &= 30\\ A &= mr\\ B &= nr \end{align}
$$\log(AB) = \log(m)+ \log(n)+2\log(r)$$
I don't know how to follow through.
The answer to the problem is $53$.
I suspect that you should have written the terms of the GP and AP as such: $$GP: 1, q, q^2, q^3, \ldots, 23, \ldots, B, \ldots$$ $$GA: 0, r, 2r, 3r, \ldots, A, \ldots, 30, \ldots$$
Where: \begin{align} q^m &= 23\\ q^n &= B\\ A &= mr\\ 30 &= nr \end{align}
Try going on from there.