Question

If a,b,c are positive real number, prove that : (i) $a^{\log b-\log c} b^{\log c-\log a} c^{\log a-\log b} = 1$

Answer 1

The identity $a^{\log b-\log c} b^{\log c-\log a} c^{\log a-\log b} = 1$ is proven to be true for all positive real numbers $a, b, c$.

Answer 2

Let $E$ be the expression. Taking $\log_k E$ and applying logarithm properties $\log(x/y) = \log x - \log y$ and $\log x^p = p \log x$, we get: $\log_k E = (\log_k b - \log_k c) \log_k a + (\log_k c - \log_k a) \log_k b + (\log_k a - \log_k b) \log_k c$. Substituting $x = \log_k a$, $y = \log_k b$, $z = \log_k c$, we have $\log_k E = (y-z)x + (z-x)y + (x-y)z$. Expanding and simplifying, $\log_k E = xy - xz + yz - yx + zx - zy = 0$. Therefore, $E = k^0 = 1$.