Question

expand \\( \log_{2} \frac{3x^2(x + 2)}{(x - 3)} \\)

Explanation:

Step1: Apply Quotient Rule

The quotient rule for logarithms is $\log_b \frac{M}{N} = \log_b M - \log_b N$. So, $\log_2 \frac{3x^2(x + 2)}{(x - 3)} = \log_2 [3x^2(x + 2)] - \log_2 (x - 3)$.

Step2: Apply Product Rule

The product rule for logarithms is $\log_b (MN) = \log_b M + \log_b N$. Applying this to $\log_2 [3x^2(x + 2)]$, we get $\log_2 3 + \log_2 x^2 + \log_2 (x + 2) - \log_2 (x - 3)$.

Step3: Apply Power Rule

The power rule for logarithms is $\log_b M^n = n\log_b M$. Applying this to $\log_2 x^2$, we get $\log_2 3 + 2\log_2 x + \log_2 (x + 2) - \log_2 (x - 3)$.

Answer:

$\log_2 3 + 2\log_2 x + \log_2 (x + 2) - \log_2 (x - 3)$