Question

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

Answer 1

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.${log\ }m^{n} = \ n\ log\ m$