Question

which expression is equivalent to \\(\log_{12} \frac{x^4 \sqrt{x^3 - 2}}{(x + 1)^5}\\)?

• \\(4\log_{12}x + \frac{1}{2}\log_{12}(x^3 - 2) - 5\log_{12}(x + 1)\\)
• \\(4\log_{12}x + \frac{1}{2}\log_{12}\frac{x^3}{2} - 5\log_{12}x + \log_{12}1\\)
• \\(\log_{12}4x + \frac{1}{2}\log_{12}(x^3 - 2) - 5\log_{12}(x) + 1\\)
• \\(4\log_{12}x + \frac{1}{2}\log_{12}(x^3 - 2) - 5\log_{12}(x + 1)\\)

Explanation:

Step1: Apply log quotient rule

The logarithm of a quotient is the difference of the logarithms: $\log_b \frac{M}{N} = \log_b M - \log_b N$. So, $\log_{12} \frac{x^4 \sqrt{x^3 - 2}}{(x + 1)^5} = \log_{12}(x^4 \sqrt{x^3 - 2}) - \log_{12}(x + 1)^5$.

Step2: Apply log product rule

The logarithm of a product is the sum of the logarithms: $\log_b (MN) = \log_b M + \log_b N$. So, $\log_{12}(x^4 \sqrt{x^3 - 2}) = \log_{12}x^4 + \log_{12}\sqrt{x^3 - 2}$.

Step3: Apply log power rule

The logarithm of a power is the exponent times the logarithm: $\log_b M^n = n\log_b M$. For $\log_{12}x^4$, we get $4\log_{12}x$. For $\log_{12}\sqrt{x^3 - 2}$, rewrite $\sqrt{x^3 - 2}$ as $(x^3 - 2)^{\frac{1}{2}}$, so $\log_{12}(x^3 - 2)^{\frac{1}{2}} = \frac{1}{2}\log_{12}(x^3 - 2)$. For $\log_{12}(x + 1)^5$, we get $5\log_{12}(x + 1)$.

Step4: Combine the results

Putting it all together: $4\log_{12}x + \frac{1}{2}\log_{12}(x^3 - 2) - 5\log_{12}(x + 1)$.

Answer:

4$\log_{12}x + \frac{1}{2}\log_{12}(x^3 - 2) - 5\log_{12}(x + 1)$ (the fourth option: 4$\log_{12}x+\frac{1}{2}\log_{12}(x^3 - 2)-5\log_{12}(x + 1)$)