Question

In a right-angled triangle, $a$ and $b$ are the lengths of the two sides and $c$ is the length of the hypotenuse. If $c+b$ and $c-b$ are numbers other than $1$, then ${\log }_{c+b}a+{\log }_{c-b}a=$

• A. ${\log }_{c-b}a\cdot {\log }_{c+b}a$
• B. $2{\log }_{c-b}a\cdot {\log }_{c+b}a$
• C. $3{\log }_{c-b}a\cdot {\log }_{c+b}a$
• D. ${\log }_{(c^2-b^2)}a$

Answer 1

The correct option is B $2{\log }_{c-b}a\cdot {\log }_{c+b}a$

$c^2=a^2+b^2\Rightarrow c^2-b^2=a^2$
Now,
${\log }_{c+b}a+{\log }_{c-b}a$
$=\frac{1}{{\log }_a(c+b)}+\frac{1}{{\log }_a(c-b)}$
$=\frac{{\log }_a(c^2-b^2)}{{\log }_a(c+b)\cdot {\log }_a(c-b)}$
$=\frac{{\log }_a\left(a^2\right)}{{\log }_a(c+b)\cdot {\log }_a(c-b)}$
$=2{\log }_{c-b}a\cdot {\log }_{c+b}a$