If ${\log _e}\left( {{{a + b} \over 2}} \right) = {1 \over 2}({\log _e}a + {\log _e}b)$, then relation between $a$ and $b$ will be
$a = b$
$a = {b \over 2}$
$a =2 b$
$a = {b \over 3}$
If ${1 \over 2} \le {\log _{0.1}}x \le 2$ then
The number of solution of ${\log _2}(x + 5) = 6 - x$ is
If ${a^x} = b,{b^y} = c,{c^z} = a,$ then value of $xyz$ is
If $3^x=4^{x-1}$, then $x=$
$(A)$ $\frac{2 \log _3 2}{2 \log _3 2-1}$ $(B)$ $\frac{2}{2-\log _2 3}$ $(C)$ $\frac{1}{1-\log _4 3}$ $(D)$ $\frac{2 \log _2 3}{2 \log _2 3-1}$
If ${a^2} + 4{b^2} = 12ab,$ then $\log (a + 2b)$ is