solve the following logarithmic equation. be sure to reject any value of x that is not in the domain of the…

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 answer.\\(\\log_{3}(x + 3) = -2\\)\\nrewrite the given equation without logarithms. do not solve for x\\n\\(3^{-2} = x + 3\\)\\nsolve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\n\\(\\boldsymbol{\\text{a.}}\\) the solution set is \\(\\boxed{}\\)\\n\\(\\quad\\) (type an exact answer in simplified form. use integers or fractions for any numbers in the expression.)\\n\\(\\boldsymbol{\\text{b.}}\\) there are infinitely many solutions.\\n\\(\\boldsymbol{\\text{c.}}\\) there is no solution.

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 answer.\\(\\log_{3}(x + 3) = -2\\)\\nrewrite the given equation without logarithms. do not solve for x\\n\\(3^{-2} = x + 3\\)\\nsolve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\n\\(\\boldsymbol{\\text{a.}}\\) the solution set is \\(\\boxed{}\\)\\n\\(\\quad\\) (type an exact answer in simplified form. use integers or fractions for any numbers in the expression.)\\n\\(\\boldsymbol{\\text{b.}}\\) there are infinitely many solutions.\\n\\(\\boldsymbol{\\text{c.}}\\) there is no solution.

Answer

Explanation:

Step1: Recall logarithm definition

For ( \log_b a = c ), we can rewrite it as ( b^c = a ). Here, ( b = 3 ), ( c = -2 ), and ( a = x + 3 ). So we get ( 3^{-2}=x + 3 ).

Step2: Simplify ( 3^{-2} )

Using the negative exponent rule ( a^{-n}=\frac{1}{a^n} ), ( 3^{-2}=\frac{1}{3^2}=\frac{1}{9} ).

Step3: Solve for ( x )

Substitute ( 3^{-2}=\frac{1}{9} ) into ( 3^{-2}=x + 3 ), we have ( \frac{1}{9}=x + 3 ). Subtract 3 from both sides: ( x=\frac{1}{9}-3=\frac{1 - 27}{9}=-\frac{26}{9} ).

Step4: Check domain

The argument of ( \log_3(x + 3) ) must be positive. So ( x+3>0 ). Substitute ( x = -\frac{26}{9} ), we get ( -\frac{26}{9}+3=\frac{-26 + 27}{9}=\frac{1}{9}>0 ), which is valid.

Answer:

The solution set is (\left{-\frac{26}{9}\right})