Solve for \( x \) : \( \frac{9}{8} \log _{4} x=-2 \) \( x=\square \) You may enter the exact value or round to 4 significant decimal places. Question Help: \( \square \) Video

To solve for \( x \) in the equation \( \frac{9}{8} \log _{4} x = -2 \), we first isolate \( \log _{4} x \): \[ \log _{4} x = -2 \cdot \frac{8}{9} = -\frac{16}{9} \] Next, we can convert the logarithmic expression to its exponential form: \[ x = 4^{-\frac{16}{9}} \] Since \( 4 \) can be expressed as \( 2^2 \): \[ x = (2^2)^{-\frac{16}{9}} = 2^{-\frac{32}{9}} \] Now, calculating this gives: \[ x = \frac{1}{2^{\frac{32}{9}}} \approx 0.0005 \] Thus, the final answer is \[ x \approx 0.0005 \quad \text{(rounded to four significant decimal places)}. \]