Problem 19. (1 point)
For the logarithm expression
\[
\log \left(y^{3} \cdot \sqrt[3]{x^{3}+1}\right)
\]
an equivalent expression is $? \quad \vec{v}$.
A. $3 \log (y)+\log (x)$
B. $3 \log (y)+3 \log \left(x^{3}+1\right)$
C. $3 \log (y)+\frac{1}{3} \log \left(x^{3}+1\right)$
D. $3 \log (y)+\log (x+1)$
Answer
Final Answer: \(\boxed{C. 3 \log (y)+\frac{1}{3} \log \left(x^{3}+1\right)}\)
Step 1 ：Break down the logarithm of the product into the sum of the logarithms: \(\log \left(y^{3}\right) + \log \left(\sqrt[3]{x^{3}+1}\right)\)
Step 2 ：Use the property of logarithms that allows us to bring the exponent out in front: \(3 \log \left(y\right) + \log \left(\sqrt[3]{x^{3}+1}\right)\)
Step 3 ：Express the cube root as a power of 1/3: \(3 \log \left(y\right) + \frac{1}{3} \log \left(x^{3}+1\right)\)
Step 4 ：Final Answer: \(\boxed{C. 3 \log (y)+\frac{1}{3} \log \left(x^{3}+1\right)}\)
