Problem 4 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$5^{x}=625$$

Step-by-Step Solution

Answer

The solution to the exponential equation \(5^{x}=625\) is \(x = 4\).

Step 1: Express both sides as a power

Express \(5^{x}\) and \(625\) as powers of a common base. The left-hand side is already in the correct form, being \(5^{x}\). In this case, we know that 625 can be expressed as \(5^{4}\), because \(5*5*5*5 = 625\). This gives us the equation \(5^{x} = 5^{4}\).

Step 2: Equate the exponents

With both sides of the equation expressed as powers of \(5\), we can now equate the exponents. This is justified because if \(a^{b} = a^{c}\) then \(b=c\) where \(a\) is any positive real number except 1. This gives us the equation \(x = 4\).

Step 3: Find the solution

The equation \(x = 4\) is the solution to the original exponential equation. Absolutely no additional operations are needed. Our solution is \(x = 4\).