Question

What does the number ${{\log }_{2}}7$ represent?
A.An integer
B.A rational number
C.An irrational number
D.A prime number

Answer 1

Here we want to find what does the number ${{\log }_{2}}7$ represents. So, assume ${{\log }_{2}}7=\dfrac{a}{b}$ and apply the properties of logarithm.

Complete step-by-step answer:
Now let us assume ${{\log }_{2}}7=\dfrac{a}{b}$, where $a$ and $b$ are integers and $b\ne 0$.
We know the property of logarithm that ${{\log }_{b}}a=\dfrac{\log a}{\log b}$.
So applying above property in ${{\log }_{2}}7$ we get,
$\dfrac{\log 7}{\log 2}=\dfrac{a}{b}$
Now let us cross multiply we get,
$b\log 7=a\log 2$ …………… (1)
Here we also know the property that $a\log b=\log {{b}^{a}}$.
Applying above property in equation (1) we get,
$\log {{7}^{b}}=\log {{2}^{a}}$
If $\log a=\log b$then $a=b$.
So in $\log {{7}^{b}}=\log {{2}^{a}}$ we get,
${{7}^{b}}={{2}^{a}}$
Since ${{2}^{a}}$ is even for all integers and ${{7}^{b}}$ is odd for all integers and no integer can be both even and odd.
Therefore, this is a contradiction and ${{\log }_{2}}7$ is an irrational number.
The correct answer is option C.

Additional information:
Integers are the numbers which can be positive, negative or zero. These numbers are used to perform various arithmetic calculations, like addition, subtraction, multiplication and division. The word integer originated from the Latin word “Integer” which means whole. It is a special set of whole numbers composed of zero, positive numbers and negative numbers and denoted by the letter Z. Rationals can be either positive, negative or zero. While specifying a negative rational number, the negative sign is either in front or with the numerator of the number, which is the standard mathematical notation. An irrational number cannot be written as a simple fraction but can be represented with a decimal. It has endless non-repeating digits after the decimal point.

Note: Here we have assume ${{\log }_{2}}7=\dfrac{a}{b}$, where $a$ and $b$ are integers and $b\ne 0$. Also, while simplifying we have used the properties of logarithm such as ${{\log }_{b}}a=\dfrac{\log a}{\log b}$, $a\log b=\log {{b}^{a}}$. Remember basic properties of logarithm.