Answer
$\frac{5}{2}$
Work Step by Step
RECALL:
(1) $\log_a{(a^k)} = k$
(2) $\sqrt{a} = a^{\frac{1}{2}}$
(3) $(a^m)^n=a^{mn}$
Use rule (2) above to obtain:
$\log_3{\sqrt{243}}= \log_3{(243^{\frac{1}{2}})}$
Note that $243=3^5$.
Thus, the expression above is equivalent to:
$\log_3{(243^{\frac{1}{2}})} = \log_3{[(3^5)^{\frac{1}{2}}]}$
Use rule (3) above to obtain:
$\log_3{[(3^5)^{\frac{1}{2}}]}
\\=\log_3{(3^{\frac{5}{2}\cdot 5})}
\\=\log_3{(3^{\frac{5}{2}})}$
Use rule (1) above to obtain:
$\log_3{(3^{\frac{5}{2}})}=\frac{5}{2}$
