OOOH ! I got one...it requires more than just logarithm knowledge though!
Be warned! I spent a VERY LONG time on this and never got the answer
But....you asked for it....here it is:
3log(x - 2) = log(2x) - 3
3log ( x - 2) = log (2x) - 3
Note that we can write 3 as log (1000)
So we have
3 log (x - 2) = log (2x) - log (1000) and we can write
log ( x - 2)^3 = log [ (2x) / 1000]
log ( x - 2)^3 = log (x / 500) we can forget the logs and solve for
(x - 2)^3 = ( x / 500) simplify
x^3 - 6x^2 + 12x - 8 = x / 500 multiply both sides by 500
500x^3 - 3000x^2 + 6000x - 4000 = x subtract x from both sides
500x^3 - 3000x^2 + 5999x - 4000 = 0 difficult to solve this, algebraically....
WolframAlpha shows the real solution as
x ≈ 2.1629