Answer
$\log_4{(x^{\frac{25}{4}})}$
Work Step by Step
Recall:
(1) $\sqrt[m]{a}=a^{\frac{1}{m}}$
(2) $\log_a {x^n}=n\cdot \log_a {x}$.
(3) $\log_a{xy}=\log_a{x} +\log_a{y}$
(4) $\log_a{\frac{x}{y}}=\log_a{x} -\log_a{y}$
($\log_a{M}=\log_a{N} \longrightarrow M=N$.)
Using Rule(2): $3\log_4{x^2}+\frac{1}{2}\log_4{x^{\frac{1}{2}}}=\log_4{x^6}+\log_4{x^{\frac{1}{4}}}.$
Using Rule(3): $\log_4{x^6}+\log_4{x^{\frac{1}{4}}}=\log_4{(x^6\cdot x^{\frac{1}{4}})}=\log_4{(x^6\cdot x^{\frac{1}{4}})}=\log_4{(x^{\frac{25}{4}})}$
