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Lösung 3.4:2b

Aus Online Mathematik Brückenkurs 1

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If we write the equation as

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we see that x appears only in the combination ex and it is therefore appropriate to treat ex as a new unknown in the equation and then, when we have obtained the value of ex, we can calculate the corresponding value of x by simply taking the logarithm.

For clarity, we set t=ex, so that the equation can be written as

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and we solve this second-degree equation by completing the square,

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which gives

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These two roots give us two possible values for ex,

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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 171 ) and we can take the logarithm of both sides to obtain

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Note: It is a little tricky to check the answer to the original equation, so we can be satisfied with substituting t=172−12  into the equation t2+t=4,

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Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_3.4:2b“