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Solve the logarithmic equation. Be sure to reject any value of [tex]\(x\)[/tex] that is not in the domain of the original logarithmic expressions. Give the exact answer.

[tex]\[ 2 \log_3(x+3) = \log_3 9 + 2 \][/tex]

Rewrite the given equation without logarithms. Do not solve for [tex]\(x\)[/tex].

[tex]\(\square\)[/tex]


Sagot :

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].