Solve the following exponential equation. Be sure to show all the steps of your work $2^{x-1}=5^{3 x+1}$

Step 1 ：Given the exponential equation \(2^{x-1}=5^{3 x+1}\)

Step 2 ：Take the natural logarithm (ln) of both sides to bring down the exponents. This gives us \(ln(2^{x-1}) = ln(5^{3x+1})\)

Step 3 ：Using the properties of logarithms, we can simplify this to \((x-1)ln(2) = (3x+1)ln(5)\)

Step 4 ：Solving for x, we get \(x = log(10^{1/log(2/125)})\)

Step 5 ：This is the exact solution in terms of logarithms. However, it might be more useful to compute a numerical approximation of this solution.

Step 6 ：Computing the numerical approximation, we get \(x = -0.556830072355780\)

Step 7 ：Final Answer: The solution to the equation \(2^{x-1}=5^{3 x+1}\) is \(x = \boxed{-0.556830072355780}\)