To find value of $a^{\log b-\log c } b^{\log c-\log a } c^{\log a -\log b}$

Question

To find value of

$$a^{\log b -\log c } b^{\log c -\log a} c^{\log a -\log b }$$ where $a, b, c$ are positive real numbers. Now using formulas

i write this as $(\frac{b}{c})^{\log a }(\frac{c}{a})^{\log b }(\frac{a}{b})^{\log c }$

How do i proceed further from here?

Thanks

Answer 1

I assume that your formula is $a^{\ln b-\ln c} b^{\ln c-\ln a} c^{\ln a-\ln b}$. Let $x=$ this number, take $\ln$ on the both of sides, we get $\ln x=(\ln b-\ln c)\ln a+(\ln c-\ln a)\ln b+(\ln a-\ln b)\ln c=0$, then $x=1$.