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

 Apr 18, 2021
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\(2\log{(a -2b)} = \log{a} + \log{b}\\ \log((a-2b)^2)=\log(ab)\\ (a-2b)^2=ab\\ a^2-4ab+4b^2=ab\\ a^2-5ab+4b^2=0\\ \)

You can factor this polynomial like this:

\((a-4b)(a-b)=0\)

So the two solutions for a/b are:

\(a-4b=0\\a=4b\\ \boxed{\frac{a}{b}=4}\)

and

\(a-b=0\\a=b\\\boxed{\frac{a}{b}=1}\)

 Apr 18, 2021

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