Solve the exponential equation. \[ e^{-c}=2^{5 c} \]

Step 1 ：Given the exponential equation \(e^{-c}=2^{5 c}\).

Step 2 ：Take the natural logarithm (ln) on both sides of the equation to bring down the exponents.

Step 3 ：Solving for the variable c, we find that c = 0.

Step 4 ：This means that when c is 0, both sides of the equation are equal.

Step 5 ：Final Answer: The solution to the equation is \(\boxed{0}\).