To condense the logarithmic expression
\[ \frac{1}{3} \left[ 2 \log(x+3) + 3 \log x - \log(x^3 + 1) \right], \]
we can apply the properties of logarithms.
-
Use the Power Rule: \( n \log a = \log(a^n) \).
- \( 2 \log(x+3) = \log((x+3)^2) \)
- \( 3 \log x = \log(x^3) \)
This modifies our expression to:
\[ \frac{1}{3} \left[ \log((x+3)^2) + \log(x^3) - \log(x^3 + 1) \right]. \]
- Use the Product Rule: \( \log a + \log b = \log(ab) \).
Combining \( \log((x+3)^2) \) and \( \log(x^3) \):
\[ \frac{1}{3} \left[ \log((x+3)^2 \cdot x^3) - \log(x^3 + 1) \right]. \]
- Use the Quotient Rule: \( \log a - \log b = \log\left(\frac{a}{b}\right) \).
Thus, we can write:
\[ \frac{1}{3} \log\left( \frac{(x+3)^2 \cdot x^3}{x^3 + 1} \right). \]
- Final Condensation:
This gives us the condensed logarithmic expression:
\[ \log\left( \sqrt[3]{ \frac{(x+3)^2 \cdot x^3}{x^3 + 1} } \right). \]
So, the final answer is:
\[ \log\left( \sqrt[3]{ \frac{(x+3)^2 \cdot x^3}{x^3 + 1} } \right). \]