Solve \[{{e}^{-\log {{e}^{2}}}}\].

Hint: We are given an expression based on logarithm function and the exponential function. We have to solve the expression and obtain the value for the same. The expression first needs to be reduced and then the value be obtained. We know that, \[a\log b=\log {{b}^{a}}\] and using this we will rearrange the expression a bit so that the expression can be easily simplified. While obtaining the value, we will put \[{{\log }_{10}}e=0.4342\] and then compute further to obtain the value of the given expression. Hence, we will have the required value.

Complete step by step answer:
According to the given question, we are given an expression and we are asked to solve the expression. The expression given to us is,
\[{{e}^{-\log {{e}^{2}}}}\]
We can see that the above expression has a logarithm function and an exponential function. We will be using their properties in order to solve the expression and hence obtain the value of the expression.
We know that, \[a\log b=\log {{b}^{a}}\] and we will use this to solve the expression. We can rewrite the given expression as,
\[\Rightarrow {{e}^{\log {{e}^{-2}}}}\]
Using the property again, we get,
\[\Rightarrow {{e}^{-2\log e}}\]
Now, we know that the value of \[{{\log }_{10}}e=0.4342\], applying this value in our expression, we have the value of the expression as,
\[\Rightarrow {{e}^{-2\left( 0.4342 \right)}}\]
We get the new expression as,
\[\Rightarrow {{e}^{-0.8486}}\]
\[\Rightarrow 0.4280\]
Therefore, the value of the given expression is \[{{e}^{-0.8486}}\] or 0.4280.

Note: The use of logarithm property should be carefully done or else the exponential terms will get unnecessarily cancelled. We also know that, \[{{e}^{\log }}=1\]. But, it should be used with the clear idea of getting the expression reduced else might create more complexity in the question.