Exponential and Logarithmic Equation Solve 2 log3xlog3 x42So

Exponential and Logarithmic Equation

Solve.

2 log₃x-log₃ (x+4)=2

Solution

2log₃x - log₃(x + 4 ) = 2 or, log₃ (x)² - log₃ (x + 4) = 2 or, log₃ [ x² /(x + 4)] = 2 Therefore, x²/ ( x + 4) = 3² = 9, ( as log₃ 3 = 1) or, x² = 9 (x + 4) = 9x + 36 or, x ² - 9x - 36 = 0

( as mlogn = log n^m and log (a/b) = log a - log b)

Then x = [ 9 ± { ( - 9)² – 4 (1)(-36)} ] / 2*1 = [ 9 ± ( 81 + 144]/2 = ( 9 ± 225) / 2 = ( 9 ± 15)/ 2 . Thus, x = either 12 or, x = - 3.

We have used the following formula for the roots of the quadratic equation ax² + bx + c = 0 ; x = [ - b ± ( b² -4ac)] / 2a