How do you simplify \[{{e}^{2\ln 4}}\]?

Hint: We will be using properties of logarithms and exponents as well to simplify the given expression. We will first take a variable say ‘y’ and equal it to the expression given. Then we will be using the property \[\ln ({{a}^{b}})=b\ln a\]. We continue solving the expression and using \[\ln (e)=1\]. And hence, we get the simplified form as a constant.

Complete step by step solution:
According to the given question, we have been given an expression to simplify. For the simplification of this expression, we will be using the properties of logarithms and exponents.
We will begin our solution by taking a variable say ‘y’ and equate it to our expression and we have,
\[y={{e}^{2\ln 4}}\]-----(1)
Now, we will take logarithm function on both sides of the equality and we get,
\[\Rightarrow \ln y=2\ln 4\]-----(2)
We know that, natural log of an exponent is 1, that is, \[\ln (e)=1\],hence we obtained the equation (2),
Next, we will be using a common property of logarithm function, that is, \[\ln ({{a}^{b}})=b\ln a\], here we will be using the reverse form of the formula which is \[b\ln a=\ln ({{a}^{b}})\]. Applying this formula in equation (2), we get,
\[\Rightarrow \ln y=\ln {{4}^{2}}\]-----(3)
Now, removing logarithm from equation (3), we get,
\[\Rightarrow y={{4}^{2}}\]
And as we know, the square of 4 is 16 i.e \[{{4}^{2}}=16\]. Hence, we get the value ‘y’ as,
\[\Rightarrow y=16\]-----(4)
Now, comparing the equation (1) and equation (4), we conclude the expression as,
\[{{e}^{2\ln 4}}=16\]

Therefore, the simplified form of the expression \[{{e}^{2\ln 4}}=16\].

Note: The logarithm functions used in the above solution should be used carefully and should be carried out in a stepwise manner.
In equation (2), we used \[\ln (e)=1\], along with this formula we also have, \[{{e}^{{{\log }_{e}}}}=1\].
For example - \[{{a}^{{{\log }_{a}}}}=1\], here we have ‘a’ raised to a logarithm with base ‘a’ which equals to 1.