Solve for $x$ :
\[
\frac{9}{8} \log _{5} x=-3
\]
\[
x=
\]

You may enter the exact value or round to 4 significant decimal places.
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Step 1 ：The given equation is \(\frac{9}{8} \log _{5} x=-3\). To solve for $x$, we first need to isolate the logarithmic term. We can do this by multiplying both sides of the equation by \(\frac{8}{9}\). This will give us \(\log _{5} x = -\frac{8}{3}\).

Step 2 ：Next, we can convert the logarithmic equation to an exponential equation. The base of the logarithm becomes the base of the exponent, the right side of the equation becomes the exponent, and $x$ is equal to this result. This gives us \(x = 5^{-\frac{8}{3}}\).

Step 3 ：We can then calculate this value to find $x$. The solution to the equation is \(x = 0.01367980757341358\)

Step 4 ：Final Answer: The solution to the equation is \(\boxed{0.0137}\) when rounded to 4 significant decimal places.