Question

Solve:

A) $lo{g}_e(x-a)+lo{g}_{e}x=1$.

B) If for $x,y\in R^{+},In\frac{x+y}{2}=\frac{1}{2}(Inx+Iny),$ Show that $x=y$

Answer 1

a] ${\log }_e(x-a)+{\log }_{e}x=1$

$\Rightarrow {\log }_e(x-a)x=1$

$\Rightarrow (x^2-ax)=e$

$\Rightarrow x^2-ax-e=0$

$\Rightarrow x=\frac{a\pm \sqrt{a^2+4e}}{2}$

b] ${\log }_e\frac{(x+y)}{2}=\frac{1}{2}lo{g}_{e}x+lo{g}_{e}y$

$\Rightarrow {\left(\frac{x+y}{2}\right)}^2=xy$

$\Rightarrow x^2+y^2+2xy-4xy=0$

$\Rightarrow (x-y{)}^2=0$

$\Rightarrow x=y$