Use the properties of logarithms and the logarithm property of equality to solve the logarithmic equation.
\[
\log _{5}(x+15)-\log _{5}(x+3)=\log _{5} x
\]
\[
x=
\]
(Type an integer or a simplified fraction.)

Step 1 ：Simplify the equation using the logarithmic property of subtraction: \(\log _{5} \left(\frac{x+15}{x+3}\right) = \log _{5} x\)

Step 2 ：Apply the logarithmic property of equality: \(\frac{x+15}{x+3} = x\)

Step 3 ：Multiply both sides of the equation by \((x+3)\) to eliminate the denominator: \(x+15 = x(x+3)\)

Step 4 ：Expand and simplify the equation: \(x+15 = x^2 + 3x\)

Step 5 ：Solve the quadratic equation by factoring or using the quadratic formula: \(x^2 + 2x - 15 = 0\)

Step 6 ：The equation can be factored as \((x+5)(x-3) = 0\)

Step 7 ：Setting each factor equal to zero, we get \(x+5 = 0\) or \(x-3 = 0\)

Step 8 ：Solving for \(x\), we have \(x = -5\) or \(x = 3\)

Step 9 ：Check the solutions in the original equation:

Step 10 ：Checking \(x = -5\): \(\log _{5}(-5+15)-\log _{5}(-5+3)=\log _{5} (-5)\)

Step 11 ：Since logarithms are only defined for positive numbers, \(x = -5\) is not a valid solution.

Step 12 ：Checking \(x = 3\): \(\log _{5}(3+15)-\log _{5}(3+3)=\log _{5} 3\)

Step 13 ：Using the logarithmic property of division, we can simplify the equation to: \(\log _{5} \left(\frac{18}{6}\right) = \log _{5} 3\)

Step 14 ：\(\log _{5} 3 = \log _{5} 3\)

Step 15 ：Since the equation is true, \(x = 3\) is a valid solution.

Step 16 ：Therefore, the solution to the logarithmic equation is \(\boxed{3}\)