If ${\log _{10}}x + {\log _{10}}\,y = 2$ then the smallest possible value of $(x + y)$ is

If $x = {\log _a}(bc),y = {\log _b}(ca),z = {\log _c}(ab),$then which of the following is equal to $1$

Let $a , b , c$ be three distinct positive real numbers such that $(2 a)^{\log _{\varepsilon} a}=(b c)^{\log _e b}$ and $b^{\log _e 2}=a^{\log _e c}$. Then $6 a+5 b c$ is equal to $........$.

If ${1 \over {{{\log }_3}\pi }} + {1 \over {{{\log }_4}\pi }} > x,$ then $x$ be

For $y = {\log _a}x$ to be defined $'a'$ must be

If ${\log _{10}}3 = 0.477$, the number of digits in ${3^{40}}$ is