Prove that ( a)^2-( b)^2= (a b ) (a b) - Vidyadip

Question

Prove that $(\log a)^2-(\log b)^2=\log \left(\frac{a}{b}\right) \cdot \log (a b)$

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Answer

$\text { L.H.S. } $
$ =(\log a)^2-(\log b)^2$
$=(\log a+\log b)(\log a-\log b) \quad \ldots\left\{\text { using identity } m^2-n^2=(m+n)(m-n)\right\} $
$=\log (a b) \log \left(\frac{a}{b}\right) $
$ =\log \left(\frac{a}{b}\right) \cdot \log (a b)$
$ =\text { R.H.S. }$

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