Solving an equation of the form $\log _{b} a=c$ Solve for $x$. \[ \log _{x} \frac{1}{1000}=3 \] Simplify your answer as much as possible.

Step 1 ：The given equation is \(\log _{x} \frac{1}{1000}=3\).

Step 2 ：This equation is in the form \(\log _{b} a=c\). Here, \(b\) is the base, \(a\) is the argument of the logarithm, and \(c\) is the result.

Step 3 ：To solve for \(x\), we need to convert the logarithmic equation to an exponential equation. The equivalent exponential form of the given logarithmic equation is \(b^c = a\).

Step 4 ：In this case, \(x\) is the base, \(\frac{1}{1000}\) is the argument, and \(3\) is the result. So, the equivalent exponential form of the given equation is \(x^3 = \frac{1}{1000}\).

Step 5 ：Solving this equation for \(x\), we get \(x = 0.1, -0.05 - 0.0866025403784439i, -0.05 + 0.0866025403784439i\).

Step 6 ：However, in the context of logarithms, the base \(x\) must be a positive real number. Therefore, the only valid solution is \(x = 0.1\).

Step 7 ：Final Answer: The solution to the equation \(\log _{x} \frac{1}{1000}=3\) is \(x = \boxed{0.1}\).