Answer

Verified

434.7k+ views

**Hint:**Here, we have to evaluate the value of $\left( {\dfrac{1}{5}} \right)\log _2^{1024}$. We know that the value of \[\log _x^{{x^n}}\] is $n$. So, first of all we have to find the value $\log _2^{1024}$ by using the above given formula, then multiply the obtained value by $\dfrac{1}{5}$ to get the final result of the given problem.

**Complete step-by-step solution:**

Given that, evaluate the value of $\left( {\dfrac{1}{5}} \right)\log _2^{1024}$.

First of all find the value of $\log _2^{1024}$, so

Now, factorising $1024$, we can write

$1024 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = {2^{10}}$.

We can write $\log _2^{1024}$as $\log _2^{{2^{10}}}$.

Now, applying the above given formula $\log _x^{{x^n}} = n$.

We can write $\log _2^{1024} = \log _2^{{2^{10}}} = 10$.

Now, we have to evaluate the value of $\left( {\dfrac{1}{5}} \right)\log _2^{1024}$.

So, $\left( {\dfrac{1}{5}} \right)\log _2^{1024} = \dfrac{1}{5} \times 10 = 2$

**Thus, the required value of $\left( {\dfrac{1}{5}} \right)\log _2^{1024}$ is $2$.**

**Note:**Some important formulae which are used to solve the problems related to logarithm are

(1) $\log _a^x + \log _a^y = \log _a^{xy}$

This formula is used to solve the problem like find the value of $\dfrac{1}{{\log _2^n}} + \dfrac{1}{{\log _3^n}} + \dfrac{1}{{\log _4^n}}$. For solving this problem first of all apply a basic logarithmic formula $\dfrac{1}{{\log _x^a}} = \log _a^x$ then by writing $\dfrac{1}{{\log _2^n}} = \log _n^2$ and similarly write other two terms and then apply this formula to get the answer.

(2) $\log _a^x - \log _a^y = \log _a^{\dfrac{x}{y}}$

This formula is used likewise formula (1) but only difference is in spite of multiplication do divisions.

(3) $\log _a^{\sqrt[n]{x}} = \dfrac{1}{n}\log _a^x$

This formula is used to solve the problem like finding the value of $\log _2^{\sqrt[{20}]{{1024}}}$. We have to solve this as similar to the above given procedure.

(4) $\log _a^x = \dfrac{{\log _c^x}}{{\log _c^a}}$.

Sometimes, we see that somewhere $\ln x$ and $\log x$ is given without base. “ $\ln x$” is called a natural logarithm whose base is $e$ that is $\ln x = \log _e^x$. If simply $\log x$ is given then it is commonly understood that its base is $10$.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Harsha Charita was written by A Kalidasa B Vishakhadatta class 7 social science CBSE

Which are the Top 10 Largest Countries of the World?

Banabhatta wrote Harshavardhanas biography What is class 6 social science CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE