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