To solve this logarithmic equation, we will first need to use the properties of logarithms to simplify the equation. We can rewrite the equation as:

log13(3x + 4) = log13(x^2 - 4x - 14)

Next, we can use the property of logarithms that states log_a(b) = log_a(c) if and only if b = c. Therefore, we have:

3x + 4 = x^2 - 4x - 14

Now we have a quadratic equation. Let's rearrange it to set it equal to zero:

x^2 - 7x - 18 = 0

Next, we can factor the quadratic equation to find the solutions for x:

(x - 9)(x + 2) = 0

Setting each factor equal to zero gives us:

x - 9 = 0
x = 9

and

x + 2 = 0
x = -2

Therefore, the solutions for the equation log13(3x + 4) = log13(x^2 - 4x - 14) are x = 9 and x = -2.