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) = 2 \\)\
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rewrite the given equation without logarithms. do not solve for x.\
\\( 3^2 = x + 3 \\)\
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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: Recall logarithm definition

For \( \log_b a = c \), it means \( b^c = a \). Here, \( b = 3 \), \( c = 2 \), \( a = x + 3 \).

Step2: Apply the definition

Using the definition, rewrite \( \log_3(x + 3) = 2 \) as \( 3^2 = x + 3 \).

Step3: Solve for x

Calculate \( 3^2 = 9 \), so \( 9 = x + 3 \). Subtract 3 from both sides: \( x = 9 - 3 = 6 \).

Step4: Check domain

The domain of \( \log_3(x + 3) \) requires \( x + 3 > 0 \), so \( x > -3 \). \( 6 > -3 \), so it's valid.

Answer:

A. The solution set is \( \boxed{6} \)