Question

1. expand $log_{6}\frac{x}{20}$

Explanation:

Step1: Apply log quotient rule

$\log_6 \frac{x}{20} = \log_6 x - \log_6 20$

Step2: Factor 20 into prime terms

$\log_6 20 = \log_6 (4 \times 5) = \log_6 4 + \log_6 5$

Step3: Rewrite 4 as $2^2$

$\log_6 4 = \log_6 2^2 = 2\log_6 2$

Step4: Substitute back to expand fully

$\log_6 \frac{x}{20} = \log_6 x - (2\log_6 2 + \log_6 5) = \log_6 x - 2\log_6 2 - \log_6 5$

Answer:

$\log_6 x - \log_6 20$ (or fully expanded: $\log_6 x - 2\log_6 2 - \log_6 5$)