Question

Let denotes the value of ${\log }_{10}\left(\frac{ab+\sqrt{{\left(ab\right)}^2-4\left(a+b\right)}}{2}\right)+{\log }_{10}\left(\frac{ab-\sqrt{{\left(ab\right)}^2-4\left(a+b\right)}}{2}\right)$ when $a=43$ and $b=57$ and B denotes the value of the expression $\left(2^{{\log }_{6}18}\right).\left(3^{{\log }_{6}3}\right)$ . Find the value of $(A-B).$

Answer 1

$A={\log }_{10}\left(\frac{{\left(ab\right)}^2-{\left(\sqrt{{\left(ab\right)}^2-4(a+b)}\right)}^2}{4}\right)$

$A={\log }_{10}(a+b)={\log }_{10}100=2$

$B=2^{{\log }_{6}8}.3^{{\log }_{6}3}$

$B=2^{(1+{\log }_{6}3)}.3^{\left({\log }_{6}3\right)}$

$B=2.2^{{\log }_{6}3}.3^{{\log }_{6}3}=2.3^{{\log }_{6}2}.3^{{\log }_{6}3}$

$B=2.3^{{\log }_{6}6}=2.3=6$

$A-B=-4$