Answer

Verified

389.7k+ views

**Hint:**Logarithmic function is an inverse function of the exponential function. An exponential function is given as $y={{a}^{x}}$, when ‘a’ is the base and ‘x’ is the exponent. Use the properties of logarithmic functions and simplify the given expression.

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

The given expression is a logarithmic function. Let us first understand what is a logarithmic function and then understand its properties.

Logarithmic function is an inverse function of the exponential function. An exponential function is given as $y={{a}^{x}}$, when ‘a’ is the base and ‘x’ is the exponent. We read this as y is equal to a raised to the power x.

A logarithmic function for the above exponential function is given as $x={{\log }_{a}}y$. We read this equation as x is equal to log to the base ‘a’ of y.

From the above logarithmic equation we understand that the function tells us about the exponent in the corresponding exponential function.

In the expression given in the question, the base of the logarithm is 2.

One of the property of logarithm functions say that ${{\log }_{a}}{{y}^{b}}=b{{\log }_{a}}y$.

Therefore, we can write ${{\log }_{2}}\left( {{2}^{x}} \right)$ as ${{\log }_{2}}\left( {{2}^{x}} \right)=x.{{\log }_{2}}\left( 2 \right)$ …. (i)

There is another property that says that ${{\log }_{a}}(a)=1$.

Then this means that ${{\log }_{2}}(2)=1$

Substitute this value in equation (i).

This gives us that ${{\log }_{2}}\left( {{2}^{x}} \right)=x(1)=x$

Hence, we simplified the given expression and we found that ${{\log }_{2}}\left( {{2}^{x}} \right)=x$

**Note:**In this given question we simplified for the logarithm function to the base 2. And from this question we can general an equation for any base that ${{\log }_{a}}\left( {{a}^{x}} \right)=x$.Now, the good thing is that we can directly use this property whenever we come across such expression.

Recently Updated Pages

When people say No pun intended what does that mea class 8 english CBSE

Name the states which share their boundary with Indias class 9 social science CBSE

Give an account of the Northern Plains of India class 9 social science CBSE

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

Advantages and disadvantages of science

10 examples of friction in our daily life

Trending doubts

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Which are the Top 10 Largest Countries of the World?

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

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

Write a letter to the principal requesting him to grant class 10 english CBSE

10 examples of law on inertia in our daily life

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