We can simplify the equation using the properties of logarithms:

log5(x+3) + log1/5(x+4) = log5(5)
log5[(x+3)(x+4)] = log5(5)

Since the logarithms on both sides are equal, we can set the expressions inside the logarithms equal to each other:

(x+3)(x+4) = 5

Expanding the left side of the equation:

x^2 + 4x + 3x + 12 = 5
x^2 + 7x + 12 = 5
x^2 + 7x + 7 = 0

Now we have a quadratic equation that we can solve by factoring or using the quadratic formula:

(x+7)(x+1) = 0

This gives us two possible solutions:

x+7 = 0
x = -7

x+1 = 0
x = -1

So the solutions to the equation are x = -7 and x = -1.