To solve the given logarithmic equation:
[tex]\[
2 \log_3(x+3) = \log_3 9 + 2
\][/tex]
We will follow a detailed step-by-step approach:
1. Simplify the right-hand side of the equation:
[tex]\(\log_3 9\)[/tex] can be simplified using the property of logarithms that states [tex]\(\log_b(a^c) = c \log_b(a)\)[/tex]. Since [tex]\(9\)[/tex] is [tex]\(3^2\)[/tex]:
[tex]\[
\log_3 9 = \log_3 (3^2) = 2
\][/tex]
So the equation becomes:
[tex]\[
2 \log_3(x+3) = 2 + 2
\][/tex]
2. Combine the constants on the right-hand side:
[tex]\[
2 \log_3(x+3) = 4
\][/tex]
3. Use a logarithmic property to rewrite the left-hand side:
We use the property [tex]\(k \log_b(a) = \log_b(a^k)\)[/tex] to combine the logarithm on the left-hand side:
[tex]\[
2 \log_3(x+3) = \log_3((x+3)^2)
\][/tex]
Substituting this into the equation, we get:
[tex]\[
\log_3((x+3)^2) = 4
\][/tex]
4. Rewrite the entire equation without logarithms:
Use the property that if [tex]\(\log_b(a) = c\)[/tex], then [tex]\(a = b^c\)[/tex]. Therefore:
[tex]\[
(x+3)^2 = 3^4
\][/tex]
So the equation without logarithms is:
[tex]\[
(x + 3)^2 = 3^4
\][/tex]
