Answer
$"...x..."$
$log{10^3}=3$
$log{10^2}=2$
$log{10^1}=1$
$log{10^0}=0$
$log{10^{-1}}=-1$
$log{10^{-2}}=-2$
$log{10^{-3}}=-3$
$log{10^\frac{1}{2}}=\frac{1}{2}$
Work Step by Step
According to the definition of $logarithms$, it is a value in which we have to raise the $base$ to get $x$.
To fill the table, we can simply input $x$ value into $log{x}$:
(Note. If the $argument$ is equal to the $base$ then the answer for expression is exponent value of the $argument$ itself)
$x=10^3$; $log{10^3}=3$
$x=10^2$; $log{10^2}=2$
$x=10^1$; $log{10^1}=1$
$x=10^0$; $log{10^0}=0$
$x=10^{-1}$; $log{10^{-1}}=-1$
$x=10^{-2}$; $log{10^{-2}}=-2$
$x=10^{-3}$; $log{10^{-3}}=-3$
$x=10^{\frac{1}{2}}$; $log{10^\frac{1}{2}}=\frac{1}{2}$
Also note that if the $base$ is omitted it means that the $base$ is $10$.