Problem 14 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$5^{2-x}=\frac{1}{125}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\)

Step 1: Express both sides as powers of the same base

We can start by recognising that 125 is \(5^3\). So, we can change the equation to the form \(5^{2-x} = 5^{-3}\)

Step 2: Equate the exponents

Once both sides are expressed as powers of the same base, we can set their exponents equal to each other due to the property of exponential functions. Therefore we get the equation \(2 - x = -3\)

Step 3: Solve for x

The equation \(2 - x = -3\) can be re-arranged to solve for \(x\). Adding \(x\) to both sides and also adding 3 to both sides of the equation, we get \(x = 2 + 3 = 5\)