Question

The value of ${\log _2}20 \times {\log _2}80 - {\log _2}5 \times {\log _2}320$ is:
A. $5$
B. $6$
C. $7$
D. $8$
E. $10$

Answer 1

Hint:In the given question, we are required to find the value of the expression involving logarithms. So, we will use the laws of logarithms such as ${\log _x}{y^n} = n{\log _x}y$. We will simplify the expression by substituting the value of ${\log _x}x$ as one and then simplifying by using simplification rules such as BODMAS.

Complete step by step answer:
So, we have, ${\log _2}20 \times {\log _2}80 - {\log _2}5 \times {\log _2}320$. Firstly, we express the numbers $20$, $80$, $5$ and $320$ as the product of their prime factors. Hence, we get the expression as,
$ \Rightarrow {\log _2}\left( {{2^2} \times 5} \right) \times {\log _2}\left( {{2^4} \times 5} \right) - {\log _2}\left( 5 \right) \times {\log _2}\left( {{2^6} \times 5} \right)$

Now, we use the property of logarithms $\ln \left( {xy} \right) = \ln \left( x \right) + \ln \left( y \right)$. So, we get,
$ \Rightarrow \left[ {{{\log }_2}\left( {{2^2}} \right) + {{\log }_2}5} \right] \times \left[ {{{\log }_2}\left( {{2^4}} \right) + {{\log }_2}5} \right] - {\log _2}\left( 5 \right) \times \left[ {{{\log }_2}\left( {{2^6}} \right) + {{\log }_2}5} \right]$
Now, we use the property of logarithm $\log {x^y} = y\log x$. So, we get,
$ \Rightarrow \left[ {2{{\log }_2}\left( 2 \right) + {{\log }_2}5} \right] \times \left[ {4{{\log }_2}\left( 2 \right) + {{\log }_2}5} \right] - {\log _2}\left( 5 \right) \times \left[ {6{{\log }_2}\left( 2 \right) + {{\log }_2}5} \right]$

We know that the value of expression ${\log _a}a$ is equal to one. So, we get,
$ \Rightarrow \left[ {2 + {{\log }_2}5} \right] \times \left[ {4 + {{\log }_2}5} \right] - {\log _2}\left( 5 \right) \times \left[ {6 + {{\log }_2}5} \right]$
Opening up the brackets and simplifying the expression, we get,
$ \Rightarrow 8 + {\left( {{{\log }_2}5} \right)^2} + 6{\log _2}5 - {\log _2}\left( 5 \right) \times \left[ {6 + {{\log }_2}5} \right]$
$ \Rightarrow 8 + {\left( {{{\log }_2}5} \right)^2} + 6{\log _2}5 - 6{\log _2}5 - {\left( {{{\log }_2}5} \right)^2}$
Cancelling the terms with same magnitude and opposite signs, we get,
$ \Rightarrow 8$
Hence, we get the value of the expression ${\log _2}20 \times {\log _2}80 - {\log _2}5 \times {\log _2}320$ as $8$.

Therefore, option D is the correct answer.

Note:Now while applying the laws of the logarithm, we should keep in mind an important rule that is the base of the logarithm functions involved should be the same in all the calculations, as the base of both the functions in the question is the same, we can apply the logarithm laws in the given question. We must follow the order of arithmetic operations according to the BODMAS rule to solve the question. Take care of calculations as it can alter the final answer.