Problem 16 - Introductory and Intermediate Algebra for College Students 4th

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 \(7^{\frac{x-2}{6}}=\sqrt{7}\) is \(x = 5\).

Step 1: Understanding the Expression

The given equation is \(7^{\frac{x-2}{6}}=\sqrt{7}\). Here both sides are in base 7. On the right side, the square root can be written as a power of half, hence the expression will be \(7^{\frac{1}{2}}\).

Step 2: Equating the Exponents

Since the base for both sides is the same (7), we equate the exponents. So, we write \(\frac{x-2}{6}=\frac{1}{2}\). This is based on the rule that if \(a^b = a^c\), then \(b = c\) where \(a\) is not equal to 0.

Step 3: Solving for \(x\)

To solve for \(x\), multiply both sides by 6. This gives: \(x-2 = 3\). Finally, add 2 to both sides to get: \(x = 5\).