Answer
$\frac{1}{2}\log_5{x} - 2$
Work Step by Step
RECALL:
(1) $\log{(b^c)}=c \cdot \log{b}$
(2) $\log_b{(xy)} = \log_b{x} + \log_b{y}$
(3) $\log_b{(\frac{x}{y})}=\log_b{x} - \log_b{y}$
Use rule (3) above to obtain:
$=log_5{\sqrt{x}}-\log_5{25}$
Note that $\sqrt{x} = x^{\frac{1}{2}}$ and $25=5^2$.
Thus, the expression above is equivalent to:
$=\log_5{(x^{\frac{1}{2}})}-\log_5{(5^2)}$
Use rule (1) above to obtain:
$=\frac{1}{2}\log_5{x} - 2\log_5{5}$
Use the rule $\log_b{b} = 1$ to obtain:
$=\frac{1}{2}\log_5{x} - 2 \cdot 1
\\=\frac{1}{2}\log_5{x} - 2
$
