Lösung 3.4:1b
Aus Online Mathematik Brückenkurs 1
In the equation, both sides are positive because the factors \displaystyle e^{x} and \displaystyle 3^{-x} are positive regardless of the value of \displaystyle x (a positive base raised to a number always gives a positive number). We can therefore take the natural logarithm of both sides,
Using the log laws, we can divide up the products into several logarithmic terms,
and using the law \displaystyle \ln a^{b}=b\cdot \ln a, we can get rid of \displaystyle x from the exponents
Collect \displaystyle x on one side and the other terms on the other,
Take out \displaystyle x on the left-hand side and use \displaystyle \ln e=1,
Then, solve for \displaystyle x,
Note: Because \displaystyle \ln 2 < \ln 13, we can write the answer as
to indicate that \displaystyle x is negative.
