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
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