Question

solve the logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expressions. give the exact answer. 2 log₂(x + 3) = log₂4 + 2 solve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the solution set is { }. (simplify your answer. use a comma to separate answers as needed.) b. there are infinitely many solutions. c. there is no solution.

Explanation:

Step1: Apply Logarithm Power Rule

The left - hand side of the equation \(2\log_{2}(x + 3)\) can be rewritten using the power rule of logarithms, which states that \(n\log_{a}b=\log_{a}(b^{n})\). So, \(2\log_{2}(x + 3)=\log_{2}(x + 3)^{2}\). The right - hand side of the equation is \(\log_{2}4+2\). We know that \(\log_{2}4=\log_{2}(2^{2}) = 2\) (since \(\log_{a}(a^{k})=k\)). So the right - hand side becomes \(2 + 2=4\), and we can rewrite it as \(\log_{2}(2^{4})\) (because \(\log_{a}(a^{b})=b\), so if we want to express 4 as a logarithm with base 2, we have \(\log_{2}(2^{4}) = 4\)). So our equation is now \(\log_{2}(x + 3)^{2}=\log_{2}(2^{4})\).

Step2: Use Logarithm Equality Property

If \(\log_{a}M=\log_{a}N\) (where \(a>0,a
eq1,M>0,N>0\)), then \(M = N\). So from \(\log_{2}(x + 3)^{2}=\log_{2}(16)\) (since \(2^{4}=16\)), we can conclude that \((x + 3)^{2}=16\).

Step3: Solve the Quadratic Equation

Take the square root of both sides: \(x + 3=\pm4\).

• Case 1: When \(x + 3 = 4\), then \(x=4 - 3=1\).
• Case 2: When \(x + 3=-4\), then \(x=-4 - 3=-7\).

Step4: Check the Domain of the Logarithm

The original logarithmic function \(\log_{2}(x + 3)\) requires that \(x+3>0\), i.e., \(x>-3\).

• For \(x = 1\): \(1+3 = 4>0\), so \(x = 1\) is in the domain.
• For \(x=-7\): \(-7 + 3=-4<0\), so \(x=-7\) is not in the domain and we reject it.

Answer:

The solution set is \(\{1\}\)