Solve the equation. Write the solution set with the exact values places. \[ 2^{1-2 x}=5^{3 x+2} \] There is no solution, \{\} . The exact solution set is \[ x \approx \]

Step 1 ：This is an exponential equation. To solve it, we need to take the logarithm of both sides. We can use the natural logarithm (ln) for this. After taking the logarithm, we can solve for x.

Step 2 ：Let's denote the equation as \(2^{1-2x} = 5^{3x+2}\).

Step 3 ：Taking the natural logarithm of both sides, we get \(\ln(2^{1-2x}) = \ln(5^{3x+2})\).

Step 4 ：Applying the properties of logarithms, we can simplify the equation to \((1-2x)\ln(2) = (3x+2)\ln(5)\).

Step 5 ：Solving this equation for x, we get \(x = \frac{\ln\left(\frac{2}{25}\right)}{\ln(500)}\).

Step 6 ：Evaluating this expression, we get the numerical value of x as \(x \approx -0.406418008071902\).

Step 7 ：\(\boxed{x \approx -0.406418008071902}\) is the solution to the equation.