To solve the equation \( 7^{x+2} = 7^{3x-8} \), we can start by recognizing that the bases on both sides of the equation are the same (both are powers of 7). Therefore, we can set the exponents equal to each other. Here are the steps to solve the equation:
-
Set the exponents equal to each other: Since \( 7^a = 7^b \) implies \( a = b \), we can write: \[ x + 2 = 3x - 8 \]
-
Rearrange the equation to isolate \(x\): To isolate \(x\), we can rearrange the equation: \[ x + 2 - 3x = -8 \] Simplifying this gives: \[ -2x + 2 = -8 \]
-
Subtract 2 from both sides: \[ -2x = -8 - 2 \] \[ -2x = -10 \]
-
Divide by -2: \[ x = \frac{-10}{-2} = 5 \]
-
Verify the solution: To ensure that our solution is correct, we can substitute \(x = 5\) back into the original equation: \[ 7^{5+2} = 7^{3(5) - 8} \] Simplifying both sides gives: \[ 7^7 = 7^{15 - 8} \] \[ 7^7 = 7^7 \] Since both sides are equal, our solution is verified.
The final solution is: \[ \boxed{5} \]