Problem 16 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$7^{\frac{x-2}{6}}=\sqrt{7}$$

Step-by-Step Solution

Answer

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

Step 1: Rewriting the equation

Here, start by expressing the square root sign as a power i.e., \(\sqrt{7}=7^{0.5}\). This means the equation becomes \(7^{\frac{x-2}{6}}=7^{0.5}\)

Step 2: Equating the exponents

Since both sides of the equation are of the same base, you can equate the exponents to each other. So, \(\frac{x-2}{6} = 0.5\)

Step 3: Solving for x

Solving for x, you multiply each side by 6 (to eliminate the fraction on the left hand side of the equation). This results in \(x - 2 = 3\). Adding 2 to both sides of the equation gives \(x = 5\)