Answer
$\frac{1}{2}$
Work Step by Step
RECALL:
$\sqrt{a} = a^{\frac{1}{2}}, a \ge 0$
Use the rule above to obtain:
$\log_{10}{\sqrt{10}}=\log_{10}{(10^{\frac{1}{2}})}$
RECALL:
$\log_a{(a^n)} = n, a \gt0, a\ne1$
Using the rule above gives:
$\log_{10}{(10^{\frac{1}{2}})}=\frac{1}{2}$
Thus,
$\log_{10}{\sqrt{10}}=\frac{1}{2}$