Sure, let's solve the logarithmic equation step-by-step and rewrite it without logarithms.
Given equation:
[tex]\[
2 \log_3 (x + 3) = \log_3 9 + 2
\][/tex]
Step 1: Simplify the right-hand side using properties of logarithms.
The equation on the right-hand side is [tex]\(\log_3 9 + 2\)[/tex].
We know that [tex]\(9\)[/tex] can be written as [tex]\(3^2\)[/tex] and hence:
[tex]\[
\log_3 9 = \log_3 (3^2) = 2
\][/tex]
So, the equation becomes:
[tex]\[
2 \log_3 (x + 3) = 2 + 2
\][/tex]
Step 2: Simplify the right-hand side.
Now we have:
[tex]\[
2 \log_3 (x + 3) = 4
\][/tex]
Step 3: Remove the coefficient from the left side using properties of logarithms.
We know that
[tex]\[
a \log_b (c) = \log_b (c^a)
\][/tex]
So we can rewrite:
[tex]\[
2 \log_3 (x + 3) = \log_3 ((x + 3)^2)
\][/tex]
Let’s substitute this back into our equation:
[tex]\[
\log_3 ((x + 3)^2) = 4
\][/tex]
Step 4: Convert the logarithmic equation to its exponential form.
By the definition of logarithms,
[tex]\[
\log_b (A) = C \implies b^C = A
\][/tex]
Applying this property:
[tex]\[
3^4 = (x + 3)^2
\][/tex]
Final Result:
Rewriting this equation, we get:
[tex]\[
(x + 3)^2 = 3^4
\][/tex]
That is:
[tex]\[
(x + 3)^2 = 9 \cdot 3^2
\][/tex]
So the rewritten equation without logarithms is:
[tex]\[
(x + 3)^2 = 9 \cdot 3^2
\][/tex]
This is the final result in the exact form we wanted, without solving for [tex]\( x \)[/tex].
