which expression is equivalent to $\\log_w\\frac{(x^{2}-6)^{4}}{\\sqrt3{x^{2}+8}}$?\n$\\bigcirc$…

which expression is equivalent to $\\log_w\\frac{(x^{2}-6)^{4}}{\\sqrt3{x^{2}+8}}$?\n$\\bigcirc$ $4\\log_w\\frac{x^{2}}{1296}-\\frac{1}{3}\\log_w(2x + 8)$\n$\\bigcirc$ $4\\log_w(x^{2}-6)-3\\log_w(x^{2}+8)$\n$\\bigcirc$ $4\\log_w(x^{2}-6)-\\frac{1}{3}\\log_w(x^{2}+8)$\n$\\bigcirc$ $4(\\log_wx^{2}-\\frac{1}{3}\\log_w(x^{2}+8)-6)$

which expression is equivalent to $\\log_w\\frac{(x^{2}-6)^{4}}{\\sqrt3{x^{2}+8}}$?\n$\\bigcirc$ $4\\log_w\\frac{x^{2}}{1296}-\\frac{1}{3}\\log_w(2x + 8)$\n$\\bigcirc$ $4\\log_w(x^{2}-6)-3\\log_w(x^{2}+8)$\n$\\bigcirc$ $4\\log_w(x^{2}-6)-\\frac{1}{3}\\log_w(x^{2}+8)$\n$\\bigcirc$ $4(\\log_wx^{2}-\\frac{1}{3}\\log_w(x^{2}+8)-6)$

Answer

Explanation:

Step1: Apply logarithm quotient rule

According to the rule (\log_{w}\frac{a}{b}=\log_{w}a - \log_{w}b), we have (\log_{w}\frac{(x^{2}-6)^{4}}{\sqrt[3]{x^{2}+8}}=\log_{w}(x^{2}-6)^{4}-\log_{w}(x^{2}+8)^{\frac{1}{3}}).

Step2: Apply logarithm power rule

Using the rule (\log_{w}a^{n}=n\log_{w}a), for (\log_{w}(x^{2}-6)^{4}), we get (4\log_{w}(x^{2}-6)), and for (\log_{w}(x^{2}+8)^{\frac{1}{3}}), we get (\frac{1}{3}\log_{w}(x^{2}+8)).

Answer:

(4\log_{w}(x^{2}-6)-\frac{1}{3}\log_{w}(x^{2}+8)) (the third option)