Lösung 3.4:2a
Aus Online Mathematik Brückenkurs 1
The left-hand side is "2 raised to something", and therefore a positive number regardless of whatever value the exponent has. We can therefore take the log of both sides,
and use the log law \displaystyle \ln a^b = b\cdot \ln a to get the exponent \displaystyle x^2-2 as a factor on the left-hand side,
Because \displaystyle e^{0}=1, so \displaystyle \ln 1 = 0, giving
This means that x must satisfy the second-degree equation
Taking the root gives \displaystyle x=-\sqrt{2} or \displaystyle x=\sqrt{2}\,.
Note: The exercise is taken from a Finnish upper-secondary final examination from March 2007.
