Solve following  x - x  e ^{5 x + 2}  = 0 .

Solution : The primary step is to factor an x out of both terms.

DO NOT DIVIDE AN x FROM BOTH TERMS!!!!

Note as well that it is extremely tempting to "simplify" the equation by dividing an x out of both terms. Though, if you do that you'll miss a solution as we'll see.

x - x  e ^{5 x + 2}  = 0

x (1 - e^{5 x + 2} ) = 0

 Hence, it's now a little easier to deal along with.  From this we can illustrates that we get one of two possibilities.

x =0     OR

1 - e ^{5 x + 2}  = 0

The first option has nothing more to do, except notice that if we had divided both of the sides by an x we would have missed this one hence be careful.  In the second possibility we've got a little more to do. It is an equation similar to the first two which we did in this section.

e^{5 x + 2}  = 1

5x + 2 = ln 1

5x + 2 = 0

x = - 2/5

Don't forget that ln 1 = 0 !

Hence, the two solutions are x = 0 and x = - 2/5

The next equation is a more complexes (looking at least...) example similar to the previous one. As a last example let's take a look at an equation that contains two different logarithms.