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 - e5 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.
e5 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.