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_5(x + 6) = 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

Using the definition of logarithms, if $\log_b a = c$, then $b^c = a$. For $\log_5(x + 6) = 3$, we have $5^3 = x + 6$.

Step2: Calculate $5^3$

$5^3 = 125$, so the equation becomes $125 = x + 6$.

Step3: Solve for x

Subtract 6 from both sides: $x = 125 - 6 = 119$.

Step4: Check domain

The argument of the logarithm $x + 6$ must be positive. For $x = 119$, $119 + 6 = 125 > 0$, so it's in the domain.

Answer:

A. The solution set is $\boxed{119}$