Solve for $x$ : \[ \begin{array}{l} \frac{9}{8} \log _{5} x=-7 \\ x= \end{array} \]

Step 1 ：The given equation is \(\frac{9}{8} \log _{5} x=-7\).

Step 2 ：To solve for \(x\), we need to convert the logarithmic equation into an exponential equation.

Step 3 ：The base of the logarithm is 5, so we can rewrite the equation as \(5\) raised to the power of the right side of the equation equals \(x\).

Step 4 ：By doing this, we get \(x = 5^{-\frac{7}{\frac{9}{8}}}\).

Step 5 ：Solving this, we get \(x = 4.475621035299599 \times 10^{-5}\).

Step 6 ：Final Answer: The solution to the equation is \(x = \boxed{4.475621035299599 \times 10^{-5}}\).