Question

How do you evaluate \[\ln \sqrt e \] ?

Answer 1

Hint: ere in this question, we have to simplify the \[\ln \sqrt e \] . The ln represents logarithm function and it is a natural logarithm. By using the properties of logarithmic function, we simplify the given function and find the value of the function. Here e is known as exponential constant.

Complete step-by-step answer:
The given function is a logarithmic function The logarithmic function is given or represented as \[{\log _b}a\] , where b is base and a is a number. In the logarithmic functions we have two different kinds, one is a common logarithmic function where it’s base is 10 and it is represented as log. The other is the natural logarithmic function where it’s base is e and it is represented as ln.
Now consider the function \[\ln \sqrt e \]
The square root of a number can be written in the form of power. Therefore \[\sqrt e = {e^{\dfrac{1}{2}}}\] .
Therefore the function is written as
 \[ \Rightarrow \ln \sqrt e = \ln {e^{\dfrac{1}{2}}}\]
Now the above function is in the form of \[\ln {a^n}\] . We have a property relate to it and it is given as \[\ln {a^n} = n\ln a\] . By applying this property, we have
 \[ \Rightarrow \dfrac{1}{2}\ln e\]
The value of \[ \Rightarrow \ln e = 1\] . On substituting the value, we get
 \[ \Rightarrow \dfrac{1}{2}(1)\]
The any number multiplied by 1 we get the same number so we have
 \[ \Rightarrow \dfrac{1}{2}\]
Hence we have evaluated the given function and obtained the solution.
Therefore \[\ln \sqrt e = \dfrac{1}{2}\]
So, the correct answer is “$\dfrac{1}{2}$”.

Note: If the question has the word log or ln it represents the given function is a logarithmic function. As we have two types of logarithmic function one is a common logarithmic function it is given as log and its base is 10. The other is a natural logarithmic function represented as ln and its base is “e”. We must know about the properties of the logarithmic functions where property holds for both log and ln functions.