logx2log 3logx4log x solve the equationSolutionlog x 2 log

log(x+2)-log 3=log(x+4)-log x    solve the equation.

Solution

log (x + 2) - log 3 = log (x + 4) - log (x) . Therefore log (x + 2) + log x - log (x + 4) = log 3 . Then, log [ x (x+2) / (x + 4)]

= log 3 ( log ab = log a + log b ; and log a/b = log a - log b) . Therefore  x (x+2) / (x + 4) = 3 or, x (x + 2) = 3 (x + 4) or, x²+ 2x = 3x + 12 or, x² - x - 12 = 0 or x² - 4x + 3x - 12 = 0 or, x (x - 4) + 3 (x - 4) = 0 or, (x - 4) (x + 3) = 0 . Therefore, either x = 4 or, x = - 3.