If $a, b, c$ are distinct positive numbers, each different from $1$, such that $[{\log _b}a{\log _c}a - {\log _a}a] + [{\log _a}b{\log _c}b - {\log _b}b]$ $ + [{\log _a}c{\log _b}c - {\log _c}c] = 0,$ then $abc =$
$1$
$2$
$3$
None of these
If ${\log _5}a.{\log _a}x = 2,$then $x$ is equal to
$\log ab - \log |b| = $
If $\log x:\log y:\log z = (y - z)\,:\,(z - x):(x - y)$ then
If ${\log _{10}}2 = 0.30103,{\log _{10}}3 = 0.47712,$ the number of digits in ${3^{12}} \times {2^8} $ is
The value of $(0.16)^{\log _{2.5}\left(\frac{1}{3}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\ldots . to \infty\right)}$ is equal to