if \\(\\log_{3}\\left(x^{8}\\sqrt3{y^{20}}\\right) = a\\log_{3}x + b\\log_{3}y\\) then\\(a =\\)\\(b…

if \\(\\log_{3}\\left(x^{8}\\sqrt3{y^{20}}\\right) = a\\log_{3}x + b\\log_{3}y\\) then\\(a =\\)\\(b =\\)question help: \\(\\boxed{\\text{video}}\\)

if \\(\\log_{3}\\left(x^{8}\\sqrt3{y^{20}}\\right) = a\\log_{3}x + b\\log_{3}y\\) then\\(a =\\)\\(b =\\)question help: \\(\\boxed{\\text{video}}\\)

Answer

Explanation:

Step1: Apply log product rule

$\log_3\left(x^8 \sqrt[3]{y^{20}}\right) = \log_3 x^8 + \log_3 \sqrt[3]{y^{20}}$

Step2: Rewrite root as exponent

$\log_3 \sqrt[3]{y^{20}} = \log_3 y^{\frac{20}{3}}$

Step3: Apply log power rule

$\log_3 x^8 = 8\log_3 x$, $\log_3 y^{\frac{20}{3}} = \frac{20}{3}\log_3 y$

Step4: Match to given form

Compare $8\log_3 x + \frac{20}{3}\log_3 y$ with $A\log_3 x + B\log_3 y$

Answer:

$A = 8$ $B = \frac{20}{3}$