To express the logarithm $\log \left[\frac{x(x+6)}{(x+5)^3}\right]$ as a sum and/or difference of logs, we use the logarithmic properties:

1. Logarithm of a product: $\log(ab) = \log(a) + \log(b)$
2. Logarithm of a quotient: $\log\left(\frac{a}{b}\right) = \log(a) - \log(b)$
3. Logarithm of a power: $\log(a^n) = n\log(a)$

Now, let's apply these rules step by step:

$\log \left[\frac{x(x+6)}{(x+5)^3}\right]$

Step 1: Apply the quotient rule

$\log \left[\frac{x(x+6)}{(x+5)^3}\right] = \log(x(x+6)) - \log((x+5)^3)$

Step 2: Apply the product rule to the numerator

$\log(x(x+6)) = \log(x) + \log(x+6)$ Thus, the expression becomes: $\log(x) + \log(x+6) - \log((x+5)^3)$

Step 3: Apply the power rule to the denominator

$\log((x+5)^3) = 3\log(x+5)$ Now the full expression is: $\log(x) + \log(x+6) - 3\log(x+5)$

Thus, the expression $\log \left[\frac{x(x+6)}{(x+5)^3}\right]$ can be written as: $\log(x) + \log(x+6) - 3\log(x+5)$

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Do you want more details or have any questions about the process?

Here are 5 related questions for practice:

1. Expand $\log \left[\frac{y(y+2)}{(y+3)^4}\right]$ as a sum and/or difference of logs.
2. Rewrite $\log \left(\frac{2x^3}{5x+1}\right)$ using logarithmic properties.
3. Simplify $\log \left(\frac{a^2b^3}{c^4}\right)$ as a sum and/or difference of logs.
4. Use properties of logarithms to simplify $\log \left(\frac{z}{z^2+1}\right)$.
5. Expand $\log \left[\frac{(x+1)(x-2)}{(x^2-4)^2}\right]$.

Tip: Always remember to apply the logarithm properties carefully in the correct order: quotient, product, and power rules!