write: \log _{\:}\left[\frac{x\left(x+6\right)}{\left(x+5\right)^3}\right] as a sum and/or difference of logs

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!

Learn how to express \log \left[\frac{x(x+6)}{(x+5)^3}\right] as a sum and/or difference of logs. This detailed solution covers the key logarithmic properties such as the product, quotient, and power rules. Ideal for students in Grades 9-12 studying logarithms in algebra.

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