Solve the equation `x^(log_(x)(x+3)^(2))=16`.

Explanation

To solve the equation \( x^{\log_{x}(x+3)^{2}} = 16 \), we can follow these steps:

### Step 1: Simplify the equation using properties of logarithms
We know that \( a^{\log_{a}(b)} = b \). Therefore, we can rewrite the left-hand side of the equation:

\[ x^{\log_{x}(x+3)^{2}} = (x+3)^{2} \]

This gives us the equation:

\[ (x + 3)^{2} = 16 \]

### Step 2: Rewrite 16 as a square
We can express 16 as \( 4^{2} \):

\[ (x + 3)^{2} = 4^{2} \]

### Step 3: Set up the difference of squares
Now, we can apply the difference of squares identity \( a^{2} - b^{2} = (a - b)(a + b) \):

\[ (x + 3)^{2} - 4^{2} = 0 \]

This leads to:

\[ (x + 3 - 4)(x + 3 + 4) = 0 \]

### Step 4: Solve the resulting equations
Now we can solve the two factors:

1. \( x + 3 - 4 = 0 \) leads to:
\[ x - 1 = 0 \implies x = 1 \]

2. \( x + 3 + 4 = 0 \) leads to:
\[ x + 7 = 0 \implies x = -7 \]

### Step 5: Check the validity of the solutions
We need to check if the solutions are valid in the context of logarithms. The base of a logarithm must be positive and cannot be equal to 1.

- For \( x = 1 \):
- The logarithm \( \log_{1}(x + 3) \) is undefined because the base cannot be 1.

- For \( x = -7 \):
- The logarithm \( \log_{-7}(x + 3) \) is also undefined because the base cannot be negative.

### Conclusion
Since both potential solutions lead to invalid logarithmic expressions, we conclude that there are no real solutions to the equation \( x^{\log_{x}(x+3)^{2}} = 16 \).

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