Questions: Solve the logarithmic equation. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. log(2x-9) = log(x+1) + log3 Rewrite the given equation without logarithms. Do not solve for x. Solve the equation to find the solution set. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Simplify your answer. Use a comma to separate answers as needed.) B. There are infinitely many solutions. C. There is no solution.

Solution Steps

To solve the logarithmic equation \(\log (2x-9) = \log (x+1) + \log 3\), we can use the properties of logarithms to combine the right-hand side into a single logarithm. Specifically, we use the property \(\log a + \log b = \log (ab)\). This allows us to rewrite the equation as \(\log (2x-9) = \log (3(x+1))\). Since the logarithms are equal, we can set the arguments equal to each other: \(2x - 9 = 3(x + 1)\). We then solve this linear equation for \(x\). Finally, we check the solution to ensure it is within the domain of the original logarithmic expressions, which requires that both \(2x-9 > 0\) and \(x+1 > 0\).

We start with the logarithmic equation: \[ \log(2x - 9) = \log(x + 1) + \log 3 \] Using the property of logarithms, we can combine the right-hand side: \[ \log(2x - 9) = \log(3(x + 1)) \]

Since the logarithms are equal, we set the arguments equal to each other: \[ 2x - 9 = 3(x + 1) \]

Expanding the right-hand side gives: \[ 2x - 9 = 3x + 3 \] Rearranging the equation leads to: \[ 2x - 3x = 3 + 9 \] \[ -x = 12 \] Thus, we find: \[ x = -12 \]

We need to check if \(x = -12\) is in the domain of the original logarithmic expressions. The conditions are:

1. \(2x - 9 > 0\)
2. \(x + 1 > 0\)

For \(x = -12\):

1. \(2(-12) - 9 = -24 - 9 = -33\) (not greater than 0)
2. \(-12 + 1 = -11\) (not greater than 0)

Since both conditions are not satisfied, \(x = -12\) is not a valid solution.

Final Answer