zmudz one year ago Find $$a/b$$ when $$2\log{(a -2b)} = \log{a} + \log{b}$$.

1. Michele_Laino

hint: I apply the properties of logarithm so I can write: ${\left( {a - 2b} \right)^2} = ab$ so, after a simplification, we have: ${a^2} + 4{b^2} - 4ab = ab$ or ${a^2} - 5ab + 4{b^2} = 0$ now I divide both sides by b^2, so I get: ${\left( {\frac{a}{b}} \right)^2} - 5\left( {\frac{a}{b}} \right) + 4 = 0$ then I make this variable change: $z = \left( {\frac{a}{b}} \right)$ so I can rewrite the last equation as follows: ${z^2} - 5z + 4$ please solve for z

2. Michele_Laino

oops.. ${z^2} - 5z + 4 = 0$