Question: Solve the following elementary exponential equation. \[ 25^{-x} = 5 \] Enter your answer (opens…

Solve the following elementary exponential equation.

\[ 25^{-x} = 5 \]

Enter your answer (opens in new window)

\[ x = \boxed{\phantom{0}} \]

Solution

To solve the exponential equation \(25^{-x} = 5\), follow these steps: Rewrite \(25\) as \(5^2\). The equation becomes: \[ (5^2)^{-x} = 5 \] Use the power of a power property to rewrite the left side: \[ 5^{-2x} = 5^1 \] Since the bases are the same, set the exponents equal: \[ -2x = 1 \] Solve for \(x\) by dividing both sides by \(-2\): \[ x = -\frac{1}{2} \] The solution is \(x = -\frac{1}{2}\).