For what values of x is log base 0.8 (x+4)> log base 0.4 (x+4) you!

We are seeking the solution of the inequality:

.

We recall that a log function  is either increasing or decreasing:

i) it is increasing if b>1, 

ii) it is decreasing if 0<b<1.

Consider the functions and .

The graphs of these functions both meet at x=1 (clearly), and after 1 they are both negative. So from 0 to 1 one of them is larger for all x, and from 1 to infinity the other is larger. (Being strictly decreasing, their graphs can only intersect once.)

We can check for a certain convenient point, for example x=0.8:

and

.

Now,  is negative since we already explained that for x>1 both functions were negative. This means that 

, and since , then this is the interval where .

So, now considering the functions and , we see that 

x+4 must be in the interval (0,1), so we solve:

0<x+4<1, which yields -4<x<-3 after we subtract by 4.

(-4, -3). Attached is the graph generated using Desmos.