Answer
423k+ views
Hint: In the provided equation of infinite exponent power, first use the power rule, which is $\ln \left( {{x^x}} \right) = x\ln x$ nd then substitute the value from the given equation in the generated equation. Then, on each of the sides, we'll use logarithm and exponential to determine the required value.
Complete step by step Answer:
Note: You should be familiar with the infinite exponential power, the power law of the logarithm, and exponential functions in order to solve these types of questions. One may become perplexed by the well-known fact that ${2^{\frac{1}{2}}} = \sqrt 2 $, or else the learner may become perplexed. The trick to solving this question is to use the logarithm on both sides.
Complete step by step Answer:
Given that the equation is ${x^{{x^{{x^ + }}}}} = 2$.
Taking logarithm in the above equation on each of the sides, we get
$\ln \left( {{x^{{x^{{x^{{x^x}}}}}}}} \right) = \ln 2$
We know that when a logarithmic term has an exponent, the logarithm power rule tells us that we can transfer the exponent to the front of the logarithm.
We will now use the power rule, that is, $\ln \left( {{x^x}} \right) = x\ln x$, in left side of the above equation.
${x^{{x^{{x^x}}}}} \cdot \ln \left( x \right) = \ln 2$
Substituting the value of ${x^{{x^{{x^x}}}}}$ in the above equation, we get
$2 \cdot \ln \left( x \right) = \ln 2$
Dividing the above equation by 2 on each of the sides, we get
$ \Rightarrow \dfrac{{2 \cdot \ln \left( x \right)}}{2} = \dfrac{{\ln 2}}{2} $
$ \Rightarrow \ln x = \dfrac{{\ln 2}}{2} $
Rearranging the right side of the above equation, we get
$ \Rightarrow \ln x = \dfrac{1}{2}\ln 2$
Using the power rule on the right side of the above equation, we get
$\ln \left( {{2^{\frac{1}{2}}}} \right) = \ln x$
Using the formula ${2^{\frac{1}{2}}} = \sqrt 2 $ in the above equation, we get
$\ln \left( {\sqrt 2 } \right) = \ln x$
Taking exponential on each of the sides in the above equation, we get
$ \Rightarrow \sqrt 2 = x $
$ \Rightarrow x = \sqrt 2 $
Thus, the given equation satisfies only when $x = \sqrt 2 $.
Hence, the correct answer is option (D).
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)