Write the expression as a sum and/or difference of logarithms. Express powers as factors.
log
[
x
(
x
+
5
)
(
x
+
3
)
4
]
,
x
>
0
 
Provide your answer below:

The given expression is:

\[
\log \left[ x (x + 5) (x + 3)^4 \right], \quad x > 0
\]

We can use the properties of logarithms to simplify it. These properties are:

1. \(\log(ab) = \log(a) + \log(b)\)
2. \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\)
3. \(\log(a^n) = n \log(a)\)

### Step 1: Apply the product rule

Using the first property, the expression can be split into a sum of logarithms:

\[
\log \left[ x (x + 5) (x + 3)^4 \right] = \log(x) + \log(x + 5) + \log\left((x + 3)^4\right)
\]

### Step 2: Apply the power rule

Now, apply the third property to the term \(\log\left((x + 3)^4\right)\), bringing the exponent 4 to the front:

\[
\log(x) + \log(x + 5) + 4 \log(x + 3)
\]

### Final Answer:

\[
\log(x) + \log(x + 5) + 4 \log(x + 3)
\]

This is the expression written as a sum of logarithms.

---

Would you like more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How would the expression change if the logarithm had a base other than 10?
2. Can you apply these logarithmic properties to an expression involving division?
3. What happens if the expression inside the logarithm contains a square root?
4. How do you handle logarithmic expressions where the base is unknown or variable?
5. Can you solve equations where logarithmic expressions are set equal to a constant?

**Tip**: Always remember the basic properties of logarithms—they're key to simplifying complex expressions!

Write the expression as a sum and/or difference of logarithms. Express powers as factors. log[x(x + 5)(x + 3)^4], x > 0

Learn how to express the logarithmic function log[x(x + 5)(x + 3)^4] as a sum and/or difference of logarithms, applying key logarithmic properties. This step-by-step guide covers product and power rules, suitable for students in Grades 9-12.

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