Problem 3 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution for \(x\) in the equation \(5^{x}=125\) is \(x = 3\).

Step 1: Expressing both sides as a power of the same base

First, recall that 125 can be expressed as \(5^3\), because \(5*5*5 = 125\). So, rewrite the equation as \(5^{x}=5^3\).

Step 2: Equating exponents

Since the bases on the left and right side of the equation are now the same, the exponents can be set equal to each other. This results in the equation \(x = 3\).

Step 3: Confirming the solution

Check if \(x = 3\) is indeed a solution of the original equation. Substitute \(x = 3\) into the original equation: \(5^3 = 125\). Since both sides of the equation are equal, the solution is verified to be correct.