Answer

Verified

412.5k+ views

**Hint:**The logarithmic function is the inverse function of the exponential function given by the formula\[{\log _b}a = c \Leftrightarrow {b^c} = \log a\], where b is the base of the logarithmic function. The logarithm is the mathematical operation that tells how many times a number or base is multiplied by itself to reach another number. There are five basic properties of the logarithm, namely Product rule, Quotient rule, Change of base rule, power rule, and equality rule.

The power rule of the logarithm is basically used to simplify the logarithm of power, rewriting it as the product of the exponent time to the logarithm base given by the formula\[{\log _a}{y^b} = b{\log _a}y\].

In this question, use the property of Power Rule of Logarithm to simplify the expression.

**Complete step by step solution:**As we can see in the given expression, we can see the base to the number 256 is 4, and for 2 the base is\[\sqrt 2 \]; hence we try to write the number in terms of the base

We know

\[

256 = {2^8} = {4^4} \\

2 = \sqrt 2 \times \sqrt 2 = {\left( {\sqrt 2 } \right)^2} \\

\]

Hence we can write the equation as

\[

A = {\log _2}{\log _2}{\log _4}256 + 2{\log _{\sqrt 2 }}2 \\

= {\log _2}{\log _2}{\log _4}{\left( 4 \right)^4} + 2{\log _{\sqrt 2 }}{\left( {\sqrt 2 } \right)^2} \\

\]

Now by using the Power Rule of the logarithm, we can write

\[

A = {\log _2}{\log _2}{\log _4}{\left( 4 \right)^4} + 2{\log _{\sqrt 2 }}{\left( {\sqrt 2 } \right)^2} \\

= {\log _2}{\log _2}4 + 4 \\

\][Since \[{\log _a} a = 1\]]

Now, again we can write \[4 = {2^2}\]

\[

A = {\log _2}{\log _2}4 + 4 \\

= {\log _2}{\log _2}{2^2} + 4 \\

= {\log _2}2 + 4 \\

\][Again using power rule]

Now use the formula \[{\log _a}a = 1\]to solve

\[

A = {\log _2}2 + 4 \\

= 1 + 4 \\

= 5 \\

\]

Hence, the value of \[A = 5\].

**Note:**Whenever log value is asked for a given number, try to bring the number in the form of the base of that logarithm by using the power rule. The logarithm of a base rule says\[{\log _a}a = 1\]as the number and the base are the same.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail