Question 29425
ln(x+2)=ln(2x-1)+3
ln(x+2)-ln(2x-1) = 3----(1)  (here the base is 10 )
log[(x+2)/(2x-1)] =3   (using log(a)-log(b) =log(a/b) )
(x+2)/(2x-1)=10^3
(x+2) = (2x-1)(10^3)
x+2 = 2X(10)^3(x)-(10^3)
x+2 = 2000x-1000
2+1000=2000x-x
1002=1999x
Therefore x = 1002/1999
Answer: x = 1002/1999
Verification: Putting  x = 1002/1999 in (1)
LHS = ln(x+2)-ln(2x-1)
=log[(1002/1999)+2] - log[(2004/1999)-1]
=log{[1002+(2X1999)]/1999}- log{[(2004-1999)]/1999}
=log{[1002+3998]/1999}- log{(5)/1999}
=log{(5000)/1999}- log{(5)/1999}
=(log5000-log1999)-(log5-log1999)
=log5000-log1999-log5+log1999
=log5000-log5
=log(5000/5)
=log(1000)
=log[(10)^3]
=3Xlog(10)
=3X1   (since log10 to the same base 10 is 1)
=3 = RHS