Question 7 Write the following expression as a single logarithm. Simplify the expression as much as possible: \( 4 \log x-\log \left(x^{2}-9\right)+2 \log (x+3) \)

To write the expression \( 4 \log x - \log (x^{2}-9) + 2 \log (x+3) \) as a single logarithm, we can use the properties of logarithms. Start by rewriting \( 4 \log x \) as \( \log(x^4) \). So now we have: \[ \log(x^4) - \log(x^2 - 9) + \log((x + 3)^2) \] Next, we combine the logarithms using the quotient and product properties: 1. Using the quotient property of logarithms, we have: \[ \log \left( \frac{x^4}{x^2 - 9} \right) \] 2. Then, adding \( \log((x + 3)^2) \) gives us: \[ \log \left( \frac{x^4 (x + 3)^2}{x^2 - 9} \right) \] Now, \( x^2 - 9 \) can be factored as \( (x - 3)(x + 3) \). Therefore, the expression simplifies to: \[ \log \left( \frac{x^4 (x + 3)^2}{(x - 3)(x + 3)} \right) \] This results in: \[ \log \left( \frac{x^4 (x + 3)}{x - 3} \right) \] Thus, the final simplified expression as a single logarithm is: \[ \log \left( \frac{x^4 (x + 3)}{x - 3} \right) \]