Question

$(log \: {a)}^{2} - ( log \: {b})^{2} = log(ab) \times log( \frac{a}{b} )$
prove that​

Answer 1

To Prove :

$\sf \log(ab) \times \log \left( \dfrac{a}{b} \right) = \left( \log a \right)^2 - \left( \log b \right)^2$

Proof :

Let's take LHS ,

$\longrightarrow \sf \log \left( ab\right) \times \log \left( \dfrac{a}{b}\right)$

We have ㏒( xy ) = ㏒ x + ㏒ y and

$\rm \log \left( \dfrac{x}{y}\right) = \log x - \log y$

$\longrightarrow \sf \left[\ \log \left( a\right) + \log \left( b\right) \right] \times \left[\ \log \left( a\right) - \log \left( b\right) \right]$

We have (a+b)(a-b) = a² -  b²

$\longrightarrow \sf \left( \log a\right)^2 - \left( \log b \right)^2$

RHS