Question

If $\log \left(\frac{a}{b-c}\right)=\log \left(\frac{b}{c-a}\right)=\log \left(\frac{c}{a-b}\right)$ then $a^a\times b^b\times c^c=1$

• A. True
• B. False

Answer 1

The correct option is A True

The question must be If $\frac{\log a}{b-c}=\frac{\log b}{c-a}=\frac{\log c}{a-b}$ then $a^a\times b^b\times c^c=1$

Solution:Let $\frac{\log a}{b-c}=\frac{\log b}{c-a}=\frac{\log c}{a-b}=k$(say)

$\Rightarrow \log a=k(b-c),\log b=k(c-a),\log c=k(a-b)$

Let $x=a^a\times b^b\times c^c$

or $\log x=a\log a+b\log b+c\log c$

or $\log x=ak(b-c)+bk(c-a)+ck(a-b)$

or $\log x=k(ab-ac+bc-ba+ca-cb)=k\times 0=0$

$\therefore \log x=0$

or $x=1$

or $x=a^a\times b^b\times c^c=1$

Hence the given statement is true.