Question

How do you evaluate \[3log{_2} 2 - \ log{_2} 4\] ?

Answer 1

Hint: In this question, we need to evaluate the given logarithmic expression. Logarithm is nothing but a power to which numbers must be raised to get some other values and also when the logarithm of a number with a base is equal to another number. Mathematically, \[\log{_b}(a)\] can be read as the logarithm of \[a\] to base \[b\]. In this question, our base is \[2\] . First, we need to make the given expression in the form of logarithmic property . Then with the help of logarithmic properties, we can easily evaluate the given expression.
Logarithmic properties used :
1.\[{log\ }m^{n} = \ n\ log\ m\]

Complete step-by-step solution:
Given, \[3log{_2}2 - \ log{_2}4\]
We can rewrite \[4\] as \[2^{2}\] ,
\[\Rightarrow \ 3log{_2}2 - \ log{_2}2^{2}\]
By using the property, \[{log\ }m^{n} = \ n\ log\ m\]
We get,
\[\Rightarrow \ 3log{_2}2 – 2log{_2}2\]
On simplifying,
We get
\[\Rightarrow \ log{_2}2\ \]
Thus \[3log{_2}2 - \ log{_2} 4\] is equal to \[\log{_2}2\] .
Final answer :
\[3log{_2}2 - \ log{_2} 4\] is equal to \[\log{_2}2\]

Note: Mathematically, there are two types of logarithm namely, common logarithm and natural logarithm. We need to know that the logarithmic function to the base \[10\] is known as the common logarithmic function and similarly the logarithmic function to the base \[e\] is known as the natural logarithmic function and it is denoted by \[\log{_e}\] . The inverse of logarithm is known as exponential. Exponential function is nothing but it is a mathematical function which is in the form of \[f\ (x)\ = \ a^{x}\] , where \[x\] is a variable and a is a constant. The most commonly used exponential base is \[e\] which is approximately equal to \[2.71828\] .
Few properties of logarithm are
1.\[log\ mn\ = \ log\ m\ + \ log\ n\]
2.\[\log\dfrac{m}{n} = \ log\ m\ \ log\ n\]
3.\[{log\ }m^{n} = \ n\ log\ m\]