Lösung 3.4:2b
Aus Online Mathematik Brückenkurs 1
If we write the equation as
we see that \displaystyle x appears only in the combination \displaystyle e^{x} and it is therefore appropriate to treat \displaystyle e^{x} as a new unknown in the equation and then, when we have obtained the value of \displaystyle e^{x}, we can calculate the corresponding value of \displaystyle x by simply taking the logarithm.
For clarity, we set \displaystyle t=e^{x}, so that the equation can be written as
and we solve this second-degree equation by completing the square,
which gives
These two roots give us two possible values for \displaystyle e^{x},
In the first case, the right-hand side is negative and because "e raised to anything" can never be negative, there is no x that can satisfy this equality. The other case, on the other hand, has a positive right-hand side (because \displaystyle \sqrt{17}>1) and we can take the logarithm of both sides to obtain
Note: It is a little tricky to check the answer to the original equation, so we can be satisfied with substituting \displaystyle t=\sqrt{17}/2-1/2 into the equation \displaystyle t^2+t=4,
