Questions: Solve the equation. Give the exact solution or solutions. log (3+x)-log (x-5)=log 5 Select one: a. -7 b. ∅ c. 3.5 d. 7

Solution Steps

To solve the equation \(\log (3+x) - \log (x-5) = \log 5\), we can use the properties of logarithms. Specifically, we can use the property that \(\log a - \log b = \log \left(\frac{a}{b}\right)\). This allows us to combine the logarithms on the left-hand side. Then, we can set the arguments of the logarithms equal to each other and solve for \(x\).

1. Use the property of logarithms to combine the left-hand side: \(\log \left(\frac{3+x}{x-5}\right) = \log 5\).
2. Since the logarithms are equal, set the arguments equal: \(\frac{3+x}{x-5} = 5\).
3. Solve the resulting equation for \(x\).

We start with the equation: \[ \log(3+x) - \log(x-5) = \log 5 \] Using the property of logarithms, we can combine the left-hand side: \[ \log\left(\frac{3+x}{x-5}\right) = \log 5 \]

Since the logarithms are equal, we can set their arguments equal to each other: \[ \frac{3+x}{x-5} = 5 \]

Cross-multiplying gives us: \[ 3 + x = 5(x - 5) \] Expanding the right side: \[ 3 + x = 5x - 25 \] Rearranging the equation: \[ 3 + 25 = 5x - x \] This simplifies to: \[ 28 = 4x \] Dividing both sides by 4: \[ x = 7 \]

Final Answer