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Chapter 12: Problem 14

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

Short Answer

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

Step by step solution

01

Express the equation with the same base

We see that we have base 5 on the left side, and \(\frac{1}{125}\) on the right side. \(\frac{1}{125}\) can be expressed as \(5^{-3}\) because \(5^{3} = 125\), and therefore \(\frac{1}{125} = 5^{-3}\). So, the equation is rewritten as: 5^{2-x} = 5^{-3}
02

Equate the powers and solve for x

Since the bases on both sides of the equation are equal, the equation becomes: 2-x = -3. Solving for x gives: x = 2 - (-3) = 5

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