Question

What is the value of \[\log \left( {\dfrac{1}{2}} \right)\] ?

Answer 1

Hint: The given function is the logarithm function; it can be defined as logarithmic functions are the inverses of exponential functions. Here we have to find the value of a given function by using quotient Properties of logarithmic and further simplify using a standard logarithm table or by standard logarithm calculator, we get the required solution.

Complete step by step answer:
The function from positive real numbers to real numbers to real numbers is defined as \[{\log _b}:{R^ + } \to R \Rightarrow {\log _b}\left( x \right) = y\], if \[{b^y} = x\], is called logarithmic function or the logarithm function is the inverse form of exponential function.

There are some basic logarithms properties
Product rule:- \[\log \left( {mn} \right) = \log m + \log n\]
Quotient rule:- \[\log \left( {\dfrac{m}{n}} \right) = \log m - \log n\]
Power rule:- \[\log \left( {{m^n}} \right) = n.\log m\]
Now, Consider the given logarithm function
 \[ \Rightarrow \,\,\log \left( {\dfrac{1}{2}} \right)\]------ (1)
Here, not mentioned any bases. Generally, \[\log \left( x \right)\] implies \[\log \] base \[10\] of \[x\], then apply quotient rule of logarithm property, then equation (1) becomes
\[ \Rightarrow \,\,\log \left( 1 \right) - \log \left( 2 \right)\]------- (2)

By the standard trigonometry table or by calculator we can find the value of common logarithms having base value 10, then
The value of \[\log \left( 1 \right) = 0\] and \[\log \left( 2 \right) = 0.3010\]
On substituting the values in equation (2), we have
\[ \Rightarrow 0 - 0.3010\]
On simplification, we get
\[ \Rightarrow \,\, - 0.3010\]

Therefore, the value of \[\log \left( {\dfrac{1}{2}} \right) = - 0.3010\].

Note:If the function contains the log term, then the function is known as logarithmic function. We have two types of logarithms namely common logarithm and natural logarithm. Remember the common logarithm represented as \[\log \] which has base 10 i.e., \[{\log _{10}}\left( x \right)\] and the natural logarithm represented as \[\ln \] which has base \[e\] i.e., \[{\ln _e}\left( x \right)\], we have a standard logarithmic property for the arithmetic operations. By using the properties, we can solve these types of questions.