Problem 8 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$5^{3 x-1}=125$$

Step-by-Step Solution

Answer

The solution to the exponential equation \(5^{3x-1}=125\) is \(x = 4/3\).

Step 1: Express the Right Side as a Power of 5

Observe that 125 is a power of 5, in fact 125 is \(5^3\). So, replace 125 with \(5^3\) in the equation. Now, the original equation \(5^{3x-1}=125\) becomes \(5^{3x-1}=5^3\)

Step 2: Equate the Exponents

Since the bases on both sides of the equation are now the same (base 5), the exponents must also be equal. So, set the exponents equal to each other: \(3x - 1 = 3\)

Step 3: Solve for 'x'

In order to solve for 'x', first add 1 to both sides to isolate '3x' on the left side of the equation. This gives: \(3x = 3 + 1\), or \(3x = 4\). Finally, divide both sides by 3 to get: \(x = 4/3\).