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(x+8) = log x + log 8 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 □ (Type an integer or a simplified fraction. 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 \(\log (x+8) = \log x + \log 8\), we can use the properties of logarithms. First, we use the property that \(\log a + \log b = \log (a \cdot b)\) to combine the right-hand side. Then, we equate the arguments of the logarithms since the logarithmic function is one-to-one. Finally, we solve the resulting algebraic equation and check if the solution is within the domain of the original logarithmic expressions.

1. Use the property of logarithms to combine the right-hand side: \(\log x + \log 8 = \log (8x)\).
2. Set the arguments of the logarithms equal to each other: \(x + 8 = 8x\).
3. Solve the resulting linear equation for \(x\).
4. Check if the solution is within the domain of the original logarithmic expressions (i.e., \(x > 0\)).

We start with the logarithmic equation: \[ \log (x + 8) = \log x + \log 8 \] Using the property of logarithms, we can combine the right-hand side: \[ \log (x + 8) = \log (8x) \]

Since the logarithmic function is one-to-one, we can set the arguments equal to each other: \[ x + 8 = 8x \]

Rearranging the equation gives: \[ 8 = 8x - x \] \[ 8 = 7x \] Dividing both sides by 7, we find: \[ x = \frac{8}{7} \]

We need to ensure that the solution is within the domain of the original logarithmic expressions. The arguments of the logarithms must be positive:

1. \(x + 8 > 0\) is satisfied for all \(x > -8\).
2. \(x > 0\) is satisfied since \(\frac{8}{7} > 0\).

Thus, the solution \(x = \frac{8}{7}\) is valid.

Final Answer