Solve for the equation:log_4 x+log_4(x+6)=2

Answer 1

Using the properties of logarithms, we can rewrite the equation as:

log_4(x(x+6)) = 2

Using the logarithmic property log_a(b) = c can be rewritten as a^c = b:

4^2 = x(x+6)

16 = x^2 + 6x

Moving all terms to one side, we get:

x^2 + 6x - 16 = 0

We can solve this quadratic equation by factoring:

(x + 8)(x - 2) = 0

Setting each factor equal to zero:

x + 8 = 0 or x - 2 = 0

x = -8 or x = 2

However, we cannot take the logarithm of a negative number, so the only valid solution is x = 2.