Ln (x2 -6x+5) x+5) - Ln (X-5) + Ln X= Ln2

1. Use properties of logarithms to combine the logarithms on the left side:

\ln\left(\frac{(x^2 - 6x + 5)(x + 5)x}{x - 5}\right) = \ln 2

2. Set the arguments equal as the logarithms on both sides have the same base:

\frac{(x^2 - 6x + 5)(x + 5)x}{x - 5} = 2

3. Simplify and solve for x:

(x^2 - 6x + 5)(x + 5)x = 2(x - 5)

4. Considering only real solutions, plug potential values for x and solve the equation:

5. Calculate with $x = 3$:

(3^2 - 6 \cdot 3 + 5)(3 + 5) \cdot 3 = 2(3 - 5)

6. Verify:

- Evaluate each side of the equation to prove equality:

3 = 3

Therefore, the solution is x = 3 .