Answer
$x=\frac{1}{3}$
Work Step by Step
RECALL:
(1) $\log_a{b} = \log_a{c} \longrightarrow b=c$
(2) $n\cdot \log_a{b} = \log_a{(b^n)}$
(3) $\log_a{a} = 1$
Use rule (2) above to obtain:
$\log_4{(x^{-2})}=\log_4{9}$
Use rule (1) above to obtain:
$x^{-2}=9$
Use the rule $x^{-m} = \frac{1}{x^m}$ to obtain:
$\frac{1}{x^2}=9$
Cross-multiply to obtain:
$9(x^2)=1(1)
\\9x^2=1
\\x^2=\frac{1}{9}
\\x^2=(\frac{1}{3})^2$
Use the rule $a^m=b^m \longrightarrow a=b$ to obtain:
$x=\frac{1}{3}$