Questions: Solve the following equation. log9(x) = 1 - log9(x-8) Select the correct choice below and, if necessary, fill in the answer box. A. The solution(s) is/are . (Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.) B. The solution is not a real number.

Solution Steps

To solve the equation \(\log_{9} x = 1 - \log_{9}(x-8)\), we can use properties of logarithms. First, we can rewrite the equation using the property that \(\log_{b} a - \log_{b} c = \log_{b} \frac{a}{c}\). This allows us to combine the logarithms on one side. Then, we can exponentiate both sides to eliminate the logarithm and solve for \(x\).

We start with the equation: \[ \log_{9} x = 1 - \log_{9}(x - 8) \] Using the property of logarithms, we can rewrite the right side: \[ \log_{9} x + \log_{9}(x - 8) = 1 \] This simplifies to: \[ \log_{9}(x(x - 8)) = 1 \]

Next, we exponentiate both sides to eliminate the logarithm: \[ x(x - 8) = 9^1 \] This simplifies to: \[ x^2 - 8x = 9 \]

Rearranging gives us the quadratic equation: \[ x^2 - 8x - 9 = 0 \] We can factor this equation or use the quadratic formula. The solutions to this equation are: \[ x = \frac{8 \pm \sqrt{(-8)^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} \] Calculating the discriminant: \[ \sqrt{64 + 36} = \sqrt{100} = 10 \] Thus, the solutions are: \[ x = \frac{8 \pm 10}{2} \] This gives us: \[ x = \frac{18}{2} = 9 \quad \text{and} \quad x = \frac{-2}{2} = -1 \]

Since \(x\) must be greater than 8 for \(\log_{9}(x - 8)\) to be defined, we discard \(x = -1\). Therefore, the only valid solution is: \[ x = 9 \]

Final Answer