## Precalculus (10th Edition)

$\log_4{(x^{\frac{25}{4}})}$
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): $\log_4{x^6}+\log_4{x^{\frac{1}{4}}}=\log_4{(x^6\cdot x^{\frac{1}{4}})}=\log_4{(x^6\cdot x^{\frac{1}{4}})}=\log_4{(x^{\frac{25}{4}})}$