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 answer. \\(\log_3(x + 3) = -3\\) solve the equation. select the correct choice below and, if necessary, fill in the answer box to complete your choice. \\(\bigcirc\\) a. the solution set is \\(\boxed{}\\) (type an exact answer in simplified form. use integers or fractions for any numbers in the expression.) \\(\bigcirc\\) b. there are infinitely many solutions. \\(\bigcirc\\) c. there is no solution.

Explanation:

Step1: Recall logarithm definition

If $\log_b(a) = c$, then $b^c = a$. Here, $b = 3$, $a = x + 3$, $c = -3$. So, $3^{-3} = x + 3$.

Step2: Simplify $3^{-3}$

$3^{-3} = \frac{1}{3^3} = \frac{1}{27}$. So, $\frac{1}{27} = x + 3$.

Step3: Solve for x

Subtract 3 from both sides: $x = \frac{1}{27} - 3 = \frac{1 - 81}{27} = -\frac{80}{27}$.

Step4: Check domain

Logarithm $\log_3(x + 3)$ requires $x + 3 > 0 \implies x > -3$. $-\frac{80}{27} \approx -2.96 > -3$, so it's in the domain.

Answer:

The solution set is $\boxed{-\dfrac{80}{27}}$