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 + 7) = 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 integer or a simplified fraction.) \\(\bigcirc\\) b there are infinitely many solutions. \\(\bigcirc\\) c there is no solution.

Explanation:

Step1: Convert log to exponential

Recall that if $\log_b(a) = c$, then $b^c = a$. For $\log_3(x + 7) = 3$, we have $3^3 = x + 7$.

Step2: Calculate $3^3$ and solve for x

$3^3 = 27$, so $27 = x + 7$. Subtract 7 from both sides: $x = 27 - 7 = 20$.

Step3: Check domain

The argument of a logarithm must be positive. For $\log_3(x + 7)$, $x + 7 > 0 \implies x > -7$. Since $20 > -7$, it is in the domain.

Answer:

The solution set is $\boxed{20}$ (corresponding to option A: The solution set is 20 (type an integer or a simplified fraction))