We are solving the equation:

$\log(3 + x) - \log(x - 2) = \log(2).$

Step 1: Apply the logarithmic subtraction rule

Using the property of logarithms: $\log(a) - \log(b) = \log\left(\frac{a}{b}\right),$ we rewrite the left-hand side: $\log\left(\frac{3 + x}{x - 2}\right) = \log(2).$

Step 2: Remove the logarithms

Since the logarithmic functions are equal, their arguments must also be equal: $\frac{3 + x}{x - 2} = 2.$

Step 3: Solve for $x$

Multiply both sides by $x - 2$ (assuming $x - 2 \neq 0$) to eliminate the fraction: $3 + x = 2(x - 2).$

Simplify the right-hand side: $3 + x = 2x - 4.$

Rearrange terms to isolate $x$: $3 + 4 = 2x - x,$ $7 = x.$

Step 4: Check for restrictions

For the logarithms to be defined:

1. $3 + x > 0 \implies x > -3$,
2. $x - 2 > 0 \implies x > 2$.

The solution $x = 7$ satisfies both conditions.

Final Answer:

$x = 7.$

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Would you like a detailed breakdown of any step or have questions? Here are some related problems you might consider:

1. Solve $\log(x+1) - \log(x-3) = \log(5)$.
2. Find $x$ in $\log(x^2) = \log(16)$.
3. Solve $\log_3(x+4) - \log_3(x) = 1$.
4. Solve $\log(x) + \log(2x - 1) = \log(15)$.
5. Simplify $\log(a) - \log(b) + \log(c)$.

Tip: Always check the domain of the logarithmic functions to ensure the solution is valid!