How do you simplify \[\ln \left( \dfrac{e}{6} \right)\]?

Hint: Now we are given with an expression in logarithm. To simplify the expression we will first use the property $\log \left( \dfrac{a}{b} \right)=\log a-\log b$ . Now we will find the value of $\ln e$ by using the definition of log function. Hence we will get a simplified expression of the given expression.

Complete step by step solution:
Now to find the value of a given expression let us first understand the concept of logarithm.
Now logarithm of a variable or a number x is written as ${{\log }_{b}}x$ where b is known as base of logarithm.
Now the function gives us the value of exponent to which the base must be raised so that we get the required number.
Now let us understand this with an example.
Suppose we have ${{\log }_{b}}x=n$ then we can write the equation in exponent form as ${{b}^{n}}=x$ .
Now consider the base b of a logarithm.
b can be any positive number which is not equal to 1.
Now if we do not have the base b mentioned in logarithm we take it as 10.
Also if we take the base of logarithm to be the constant e, then the logarithm is called natural logarithm and is denoted by ln.
Hence $\ln x$ is nothing but ${{\log }_{e}}x$ .
Now consider the given equation \[\ln \left( \dfrac{e}{6} \right)\].
Now we know that $\ln \left( \dfrac{a}{b} \right)=\ln a-\ln b$ .
Hence using this property in the given equation we get, $\ln \left( \dfrac{e}{6} \right)=\ln e-\ln 6$
Now by using the definition of log we know that $\ln e=1$ hence the given equation can be written as,
$\Rightarrow 1-\ln 6$

Hence the value of \[\ln \left( \dfrac{e}{6} \right)\] is $1-\ln 6$.

Note: Now note that in general we have ${{\ln }_{a}}a=1$ as ${{a}^{1}}=a$ . Also note that for any base the value of ${{\log }_{a}}1$ is always 0. This is because for all real numbers we have ${{a}^{0}}=1$ . The logarithm of negative numbers and zero is not defined.