Question

which shows the first step in the solution to the equation \\(\log_{2} x + \log_{2}(x - 6) = 4\\)? this is a multi - part item. solving equations using properties of logarithms using the product rule

Explanation:

Step1: Recall Logarithm Product Rule

The product rule for logarithms states that \(\log_b M + \log_b N=\log_b(MN)\) for \(b>0,b
eq1,M>0,N>0\).

Step2: Apply Product Rule to the Equation

Given the equation \(\log_2 x+\log_2(x - 6)=4\), we apply the product rule. So we combine the two logarithms on the left - hand side into a single logarithm: \(\log_2[x(x - 6)] = 4\) or \(\log_2(x(x - 6))=4\) (which can also be written as \(\log_2(x^2-6x)=4\) after expanding the product inside the logarithm).

Answer:

The first step in solving the equation \(\log_2x+\log_2(x - 6)=4\) is to use the product rule of logarithms to combine the two logarithmic terms. Applying the rule \(\log_b M+\log_b N = \log_b(MN)\), we get \(\log_2[x(x - 6)]=4\) (or \(\log_2(x^{2}-6x)=4\)).