I came across this exercise and I don't know what to do next (or if I solved it correctly to this point)

x^{log_3 (x+1)} + (x+1)^{log_3 x} = 2x^2

log_3 {x^{log_3 (x+1)}} + log_3 {(x+1)^{log_3 x}} = log_3 {(2x^2)}

2 log_3 {x} * log_3 {(x+1)} = log_3 {2} + 2log_3 {x}

You are making the mistake of assuming that if


a + b = c


then

log(a) + log(b) = log(c)


which is not right.

Rather than using brute force I suggest you try a few numbers that look like reasonable candidates. You want things to cancel out, and at first glance it would seem that trying values for x like x = 3, or x=2 may be promising. You can also try x = 1, since 1 raised to a power is always 1 and log of 1 is always 0. Post back with what you get.