Solve for b:
2 log(a - 2 b) = log(a) + log(b)
Subtract log(a) + log(b) from both sides:
-log(a) + 2 log(a - 2 b) - log(b) = 0
-log(a) + 2 log(a - 2 b) - log(b) = log(1/a) + log((a - 2 b)^2) + log(1/b) = log((a - 2 b)^2/(a b)):
log((a - 2 b)^2/(a b)) = 0
Cancel logarithms by taking exp of both sides:
(a - 2 b)^2/(a b) = 1
Multiply both sides by a b:
(a - 2 b)^2 = a b
Subtract a b from both sides:
(a - 2 b)^2 - a b = 0
Expand out terms of the left hand side:
a^2 - 5 a b + 4 b^2 = 0
The left hand side factors into a product with two terms:
(a - 4 b) (a - b) = 0
Split into two equations:
a - 4 b = 0 or a - b = 0
Subtract a from both sides:
-4 b = -a or a - b = 0
Divide both sides by -4:
b = a/4 or a - b = 0
Subtract a from both sides:
b = a/4 or -b = -a
Multiply both sides by -1:
b = a/4 So, a / b = 4