Fernandez Turner
10/01/2023 · Junior High School
Q3: Use properties of logarithms to expand the expression \( \ln \left(\frac{x^{5} \sqrt{x^{3}+2}}{(x-3)^{4}}\right) \) as much as possible.
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Tutor-Verified Answer
Step-by-step Solution
To expand the expression \( \ln \left(\frac{x^{5} \sqrt{x^{3}+2}}{(x-3)^{4}}\right) \) using properties of logarithms, we can apply the following properties:
1. \( \ln \left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
2. \( \ln(a \cdot b) = \ln(a) + \ln(b) \)
3. \( \ln(a^b) = b \cdot \ln(a) \)
4. \( \ln(\sqrt{a}) = \frac{1}{2} \ln(a) \)
Now, let's apply these properties step by step:
1. Start with the original expression:
\[
\ln \left(\frac{x^{5} \sqrt{x^{3}+2}}{(x-3)^{4}}\right)
\]
2. Use the quotient property:
\[
= \ln(x^{5} \sqrt{x^{3}+2}) - \ln((x-3)^{4})
\]
3. Now, expand \( \ln(x^{5} \sqrt{x^{3}+2}) \) using the product property:
\[
= \ln(x^{5}) + \ln(\sqrt{x^{3}+2}) - \ln((x-3)^{4})
\]
4. Next, apply the power property to each logarithm:
\[
= 5 \ln(x) + \frac{1}{2} \ln(x^{3}+2) - 4 \ln(x-3)
\]
Putting it all together, the fully expanded expression is:
\[
\ln \left(\frac{x^{5} \sqrt{x^{3}+2}}{(x-3)^{4}}\right) = 5 \ln(x) + \frac{1}{2} \ln(x^{3}+2) - 4 \ln(x-3)
\]
Quick Answer
The expanded expression is \( 5 \ln(x) + \frac{1}{2} \ln(x^{3}+2) - 4 \ln(x-3) \).
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