Question

solve the following logarithmic equation. be sure to reject any value of x that is not in the domain of the original logarithmic expression. give the exact \\(\log _{3}(x + 16) = 3\\) 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 \\(\boxed{}\\). (type an integer or a simplified fraction.) b. there are infinitely many solutions. c. there is no solution.

Explanation:

Step1: Convert log to exponential

Recall that \(\log_b a = c\) implies \(b^c = a\). For \(\log_2(x + 16) = 3\), we have \(2^3 = x + 16\).
\(2^3 = 8\), so \(8 = x + 16\).

Step2: Solve for x

Subtract 16 from both sides: \(x = 8 - 16 = -8\).

Step3: Check domain

The argument of a logarithm must be positive. For \(\log_2(x + 16)\), \(x + 16>0\). When \(x = -8\), \(-8 + 16 = 8>0\), so it's in the domain.

Answer:

A. The solution set is \(\{-8\}\)