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.\\(\log_3(x + 7) + \log_3(x + 5) = 1\\)\
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{}\\).\
(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 log product rule

Using the property \(\log_b M + \log_b N=\log_b(MN)\), we get \(\log_3[(x + 7)(x + 5)] = 1\).

Step2: Convert to exponential form

Recall that \(\log_b a = c\) implies \(b^c=a\). So, \(3^1=(x + 7)(x + 5)\), which simplifies to \(x^2+12x + 35 = 3\).

Step3: Solve quadratic equation

Rearrange to \(x^2+12x + 32 = 0\). Factor: \((x + 4)(x + 8)=0\). Solutions are \(x=-4\) or \(x=-8\).

Step4: Check domain

For \(\log_3(x + 7)\) and \(\log_3(x + 5)\), we need \(x+7>0\) (i.e., \(x>-7\)) and \(x + 5>0\) (i.e., \(x>-5\)). \(x=-8\) fails (\(-8<-7\) and \(-8<-5\)), \(x=-4\) passes (\(-4>-7\) and \(-4>-5\)).

Answer:

-4