Solve for x log6x + log6 (x + 5) = 2
To solve the equation log6x + log6(x + 5) = 2, we can combine the logarithms using the product rule, which states that loga(x) + loga(y) = loga(xy):
log6(x(x + 5)) = 2
log6(x^2 + 5x) = 2
Now, we can rewrite the logarithmic equation in exponential form:
6^2 = x^2 + 5x
36 = x^2 + 5x
Rearrange the equation into a quadratic form:
x^2 + 5x - 36 = 0
Now, we can factor the quadratic equation:
(x + 9)(x - 4) = 0
Setting each factor to zero:
x + 9 = 0 or x - 4 = 0
x = -9 or x = 4
Therefore, the solutions for x are -9 and 4.