Question

Find \(a/b\) when \(2\log{(a -2b)} = \log{a} + \log{b}\).

Answer 1

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

Answer 2

oops..
\[{z^2} - 5z + 4 = 0\]