Log5(x-1)+log5(x-5)= log5(x+2)?

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To solve this logarithmic equation, first combine the two logarithms on the left side using the product rule of logarithms:

log5((x-1)(x-5)) = log5(x+2)

Now, set the two expressions inside the logarithms equal to each other:

(x-1)(x-5) = x+2

Expand the left side of the equation:

x^2 - 5x - x + 5 = x + 2

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

Rearrange the terms to set the equation to zero:

x^2 - 6x - x + 5 - 2 = 0

x^2 - 7x + 3 = 0

Using the quadratic formula to solve for x:

x = (7 ± √(7^2 - 413)) / 2

x = (7 ± √(49 - 12)) / 2

x = (7 ± √37) / 2

Therefore, the solutions to the logarithmic equation are:

x = (7 + √37) / 2 and x = (7 - √37) / 2