solve the logarithmic equation. Please show all work. log5x+log5(x-3)=log5(x+5)

To solve the logarithmic equation log5x + log5(x - 3) = log5(x + 5), we can use the properties of logarithms to simplify the equation and then solve for x.

Step 1: Apply the product rule of logarithms
log5x + log5(x - 3) = log5(x + 5)
log5(x · (x - 3)) = log5(x + 5)

Step 2: Simplify the left-hand side of the equation
log5(x^2 - 3x) = log5(x + 5)

Step 3: Apply the one-to-one property of logarithms
x^2 - 3x = x + 5

Step 4: Rearrange the equation and set it to zero
x^2 - 3x - x - 5 = 0
x^2 - 4x - 5 = 0

Step 5: Factor the quadratic equation
(x - 5)(x + 1) = 0

Step 6: Solve for x
x - 5 = 0 or x + 1 = 0
x = 5 x = -1

Therefore, the solution to the logarithmic equation log5x + log5(x - 3) = log5(x + 5) is x = 5 or x = -1.