Question

How do you simplify \[\ln \left( {\dfrac{1}{2}} \right) - \ln \left( 1 \right)\]?

Answer 1

Hint:In the given question, we have been given an expression. It is an expression of a natural logarithm. There are two terms of the natural logarithm. They are separated by the minus sign. We have to simplify the expression. We can easily do that if we know the properties of the logarithm.

Formula Used:
We are going to use the difference formula of natural logarithm:
\[\ln a - \ln b = \ln \left( {\dfrac{a}{b}} \right)\]

Complete step by step answer:
The given expression is
\[p = \ln \left( {\dfrac{1}{2}} \right) - \ln \left( 1 \right)\]
To solve this question, we are going to use the difference formula of natural logarithm, which is,
\[\ln a - \ln b = \ln \left( {\dfrac{a}{b}} \right)\]
Substituting \[a = \dfrac{1}{2}\] and \[b = 1\], we have,
\[p = \ln \left( {\dfrac{{\dfrac{1}{2}}}{1}} \right) = \ln \left( {\dfrac{1}{2}} \right)\]
Now, \[\ln \left( {\dfrac{1}{2}} \right) = p\]
Again, applying the same formula,
\[p = \ln \left( 1 \right) - \ln \left( 2 \right)\]
Now, we know, \[\ln \left( 1 \right) = 0\]
Hence, \[p = - \ln \left( 2 \right)\]

Additional Information:
The \[\log \] function has other basic properties too:
\[{\log _x}{x^n} = n\]
\[{\log _b}a = n \Rightarrow {b^n} = a\]
\[{\log _a}b = \dfrac{1}{{{{\log }_b}a}}\]

Note: In this question, to solve for the answer, we needed to know the properties of the logarithmic function. We needed to know-how and are related. If we know such basic things about any topic, we can easily solve for the answer.