Question

Solve the following equations.
$|x-10|lo{g}_2(x-3)=2(x-10)$

Answer 1

$|x-10|lo{g}_2(x-3)=2(x-10)$

Case I: if $x\ge 10$

$(x-10)lo{g}_2(x-3)=2(x-10)$

$\Rightarrow (x-10)\left[lo{g}_2\right(x-3)-2]=0$

$\Rightarrow \therefore x=10$ or $lo{g}_2(x-3)=2$

$x=10$ or $x-3=4$

$x=10$ or $x=7$

But for $x\ge 10$ the only possibility is $x=10$

Case 2 : if $x<10$

$(10-x)lo{g}_2(x-3)=2(x-10)$

$\Rightarrow \left[lo{g}_2\right(x-3\left)\right](10-x)=2(x-10)$

$\Rightarrow (x-10)[2+lo{g}_2(x-3\left)\right]=0$

$\therefore x=10$ or $lo{g}_2(x-3)=-2$

$x=10$ or $x-3=\frac{1}{4}$

$x=10$ or $x=\frac{13}{4}$

But for $x<10;$ the only possibility is $x=13/4$

$\therefore$ the values of x satisfying the above equation is

$x=10$ and $x=13/4$