log5 (x+2)+log5(x-2)=3

To solve the equation log5 (x+2) + log5 (x-2) = 3, you can use the properties of logarithms to simplify the expression and isolate the variable x.

Step 1: Combine the logarithms using the product rule of logarithms. According to this rule, when adding logarithms with the same base, you can write the sum as the logarithm of the product:

log5 ((x+2)(x-2)) = 3

Step 2: Simplify the expression inside the logarithm:

log5 (x^2 - 4) = 3

Step 3: Rewrite the logarithmic equation in exponential form. If logb (y) = x, then by definition, b^x = y. In our case, the base is 5, the exponent is 3, and the result is (x^2 - 4):

5^3 = x^2 - 4

Step 4: Calculate 5^3 to get:

125 = x^2 - 4

Step 5: Add 4 to both sides of the equation:

125 + 4 = x^2

129 = x^2

Step 6: Take the square root of both sides of the equation:

√129 = √(x^2)

±√129 = x

So, the solution to the equation log5 (x+2) + log5 (x-2) = 3 is x = ±√129.