Question

How do I solve \[\ln \left( {{e}^{x}} \right)\] ?

Answer 1

Hint: In this question here a function \[\ln \] which is actually a logarithmic function with it’ base \[e\] where \[e\] is the exponential constant also \[\ln v\] is written as \[{{\log }_{e}}v\] is and we know the properties of \[\log \] and one of the property is \[\ln ({{a}^{b}})=b\ln (a)\] and when we apply this in given question we get only \[x\].

Complete step by step solution:
As the given function is \[\ln \left( {{e}^{x}} \right)\] as it is already solved we just need to simplify it.
Just recall the property of \[\log \] that is
\[\Rightarrow \ln ({{a}^{b}})=b\ln (a)\]
Now compare it with a given function
\[\Rightarrow a=e\text{ , b = x}\]
\[\Rightarrow \ln \left( {{e}^{x}} \right)=x\ln e\]
And \[\ln e\] can also be written as \[{{\log }_{e}}e\]
\[\Rightarrow x{{\log }_{e}}e\]
Also, we know that value of \[\log \] function with the same base value is \[1\]
\[\Rightarrow x\]

Hence the simplified value of \[\ln \left( {{e}^{x}} \right)\] is \[x\].

Note: On solving these types of questions first write down the function and look it carefully and recall the properties of that function just in this question logarithm recall the properties of the log then you will be able to solve or simplify.