## College Algebra (10th Edition)

$\log_\sqrt{2}{4} = 4$
Let $\log_\sqrt{2}{4}=y$ RECALL: $\log_a{x} = y \longrightarrow a^y=x$ Use this rule to obtain: $\log_\sqrt2{4} = y \longrightarrow (\sqrt{2})^y=4$ Write $4$ as $2^2$ to obtain: $(\sqrt{2})^y=2^2$ Note that $\sqrt{2} = 2^{\frac{1}{2}}$. Thus, the equation above is equivalent to: $(2^{\frac{1}{2}})^y = 2^2$ Use the rule $(a^m)^n=a^{mn}$ to obtain: $2^{\frac{1}{2} \cdot y} = 2^2 \\2^{\frac{y}{2}} = 2^2$ Use the rule "$a^m=a^n \longrightarrow m=n$" to obtain: $\frac{y}{2} = 2$ Multiply $2$ to both sides of the equation to obtain: $2(\frac{y}{2}) = 2(2) \\y = 4$ Thus, $\log_\sqrt{2}{4} = 4$