Question
Prove that: $(\log a)^2-(\log b)^2=\log \left(\frac{a}{b}\right) \cdot \log (a b)$
✓
Answer
$\text { L.H.S }=(\log a)^2-(\log b)^2$
$\Rightarrow \text { L.H.S }=(\log a+\log b)(\log a-\log b)$
$\Rightarrow \text { L.H.S }=\log (a b) \log \left(\frac{a}{b}\right)$
$\Rightarrow \text { L.H.S }=\log \left(\frac{a}{b}\right) \times \log (a b)$
$\Rightarrow \text { L.H.S }=\text { R.H.S }$
Hence proved.
$\Rightarrow \text { L.H.S }=(\log a+\log b)(\log a-\log b)$
$\Rightarrow \text { L.H.S }=\log (a b) \log \left(\frac{a}{b}\right)$
$\Rightarrow \text { L.H.S }=\log \left(\frac{a}{b}\right) \times \log (a b)$
$\Rightarrow \text { L.H.S }=\text { R.H.S }$
Hence proved.
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