Solve the following equation for $x$ : \[ 2^{-x}=15 \]

Step 1 ：We are given the equation \(2^{-x}=15\).

Step 2 ：This is an exponential equation. To solve for \(x\), we need to take the logarithm of both sides. We can use the natural logarithm (ln) or the base 10 logarithm (log). The choice of logarithm doesn't matter because the ratio of two logarithms is a constant. However, the natural logarithm is often easier to work with because it has nicer properties.

Step 3 ：Applying the logarithm to both sides of the equation, we get \(x = -\log_{2}{15}\).

Step 4 ：Solving this equation, we find that \(x = -3.9068905956085187\).

Step 5 ：The negative sign indicates that the solution is in the negative domain of the x-axis.

Step 6 ：Final Answer: The solution to the equation \(2^{-x}=15\) is \(x = \boxed{-3.9068905956085187}\).