What is the solution of equation log4 (x+1)+log3 (x+4)=1?

To use the product property to the left side, we'll have
to create matching bases;

log 4 (x+4) = log 3 (x+4)/log 3 4
=> log 3 (x+4) = (log 3 4)*(log 4 (x+4))

The
equation will become:

log4 (x+1)+(log 3 4)*(log 4 (x+4)) =
log 4 4

(log 3 4)*(log 4 (x+4)) = log 4 4 - log4
(x+1)

(log 3 4)*(log 4 (x+4)) = log 4
[4/(x+1)]

(ln 4/ln 3)*(log 4 (x+4)) = log 4
[4/(x+1)]

(ln 4/ln3) `~=`

log
4 (x+4)^1.263 = log 4 [4/(x+1)]

Since the bases are
matching, we'll use one to one property:

(x+4)^1.263 =
[4/(x+1)]

(x+1)*(x+4)^1.263 =
4

The factors could be:

x+ 1
=1

x = 0 => (x+4)^1.263 = 4
impossible

x+1 = -1

x = -2
=> (-2+4)^1.263 = -4 impossible

x + 1 =
2

x = 1 => (1+4)^1.263 = 2
impossible

x + 1 = -2

x = -3
=> (-3+4)^1.263 = -2 impossible

x + 1 = 4 =>
(3+4)^1.263 = 1 impossible

x =
3

x + 1=-4

x = -5 =>
(-5+4)^1.263 = -1  => x = -5

We'll
accept as solution of equation x = -5