Write the expression as a sum and/or difference of logarithms. Express powers as factors. \[ \log \left[\frac{x(x+7)}{(x+8)^{8}}\right], x>0 \]

To express the logarithmic expression as a sum and/or difference of logarithms, we can use the properties of logarithms. Starting with the expression: \[ \log \left[\frac{x(x+7)}{(x+8)^{8}}\right] \] We can break it down using the quotient rule of logarithms, which states that \(\log\left(\frac{A}{B}\right) = \log A - \log B\). This gives us: \[ \log[x(x+7)] - \log[(x+8)^{8}] \] Next, we can use the product rule of logarithms, which states that \(\log(AB) = \log A + \log B\). Hence, \[ \log[x(x+7)] = \log x + \log(x+7) \] So now, substituting back: \[ \log x + \log(x+7) - \log[(x+8)^{8}] \] Finally, applying the power rule of logarithms, which states that \(\log(A^n) = n \log A\), we have: \[ \log x + \log(x+7) - 8 \log(x+8) \] Thus, the expression can be rewritten as a sum and difference of logarithms: \[ \log x + \log(x+7) - 8 \log(x+8) \]