Condense the expression to the logarithm of a single quantity.(Assume x > 1.) 5[4 ln(x) − ln(x + 1) − ln(x − 1)]



To condense the expression \(5[4 \ln(x) - \ln(x + 1) - \ln(x - 1)]\) to the logarithm of a single quantity, we can apply the properties of logarithms.

### Step 1: Distribute the \(5\)
\[
5[4 \ln(x) - \ln(x + 1) - \ln(x - 1)] = 20 \ln(x) - 5 \ln(x + 1) - 5 \ln(x - 1)
\]

### Step 2: Apply the power rule for logarithms
Using the rule \(a \ln(b) = \ln(b^a)\), we rewrite each term:
\[
= \ln(x^{20}) - \ln((x + 1)^5) - \ln((x - 1)^5)
\]

### Step 3: Apply the quotient rule for logarithms
Using the rule \(\ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\), we combine the terms into a single logarithm:
\[
= \ln\left(\frac{x^{20}}{(x + 1)^5 (x - 1)^5}\right)
\]

### Final Answer
\[
\ln\left(\frac{x^{20}}{(x + 1)^5 (x - 1)^5}\right)
\]

Would you like a further breakdown or have any questions?

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Here are some additional related questions to consider:
1. What are the steps to expand a condensed logarithmic expression?
2. How would this expression change if we assumed \(x < 1\)?
3. What is the significance of each logarithmic property used here?
4. How does this approach help in solving logarithmic equations?
5. What alternative methods could simplify this expression?

**Tip:** Always remember to apply the power rule first before combining terms using the quotient rule for efficient simplification.

Condense the expression to the logarithm of a single quantity. (Assume x > 1.) 5[4 ln(x) − ln(x + 1) − ln(x − 1)]

Learn how to condense the expression 5[4 ln(x) − ln(x + 1) − ln(x − 1)] to a single logarithm using logarithmic properties. This step-by-step guide covers the distributive property, power rule, and quotient rule, suitable for students in Grades 9-11.

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