Better late than never:
This isn't really a coding problem, but here's the method anyway. From the original equation we know a couple facts about A and h. First, we know that A > 0 (strictly greater than) because ln is undefined for values of A <= 0. For the same reason we know that h > 0. However, we also know that 1 - h > 0, which implies h < 1 (again, strictly). This gives us A > 0 and 0 < h < 1.
Now, we first simplify the equation by combining the logs on the right side. ln(a) - ln(b) = ln(a / b) is a logarithmic identity, so we get ln(A) = ln(h / (1 - h) ).
Now, since the exponential function is continuous over the real numbers we can take the exponential of both sides to remove the logs and get A = h / (1 - h).
From here on out it's just algebra, but for the sake of completeness I'll go step by step. Since we know both A and h are strictly greater than 0 we can take the reciprocal of both sides to give us 1 / A = (1 - h) / h. We then split up the fraction on the right side to get 1 / A = (1 / h) - 1. Add 1 to give us 1 + (1 / A) = 1 / h --------> (A + 1) / A = 1 / h. Again, A + 1 > 0, so take the reciprocal again to give us A / (A + 1) = h QED.