log6(𝑥 + 4)+ log6(𝑥 − 2)= log6 4𝑥 Find 𝑥

Step 1: Use the property of logarithms that states logb(m) + logb(n) = logb(mn) to combine the left side of the equation. This gives us:

log6[(x + 4)(x - 2)] = log6(4x)

Step 2: Since the bases of the logarithms are the same, we can set the arguments equal to each other:

(x + 4)(x - 2) = 4x

Step 3: Expand the left side of the equation:

x^2 + 2x - 8 = 4x

Step 4: Rearrange the equation to set it equal to zero:

x^2 - 2x - 8 = 0

Step 5: Factor the quadratic equation:

(x - 4)(x + 2) = 0

Step 6: Set each factor equal to zero and solve for x:

x - 4 = 0 or x + 2 = 0

x = 4 or x = -2

However, if we substitute x = -2 back into the original equation, we get a negative number inside the logarithm, which is undefined. Therefore, the only solution is x = 4.