
The equation ${x^{{x^{{x^ + }}}}} = 2$ is satisfied when $x$ is equal toA. InfinityB. 2C. $\sqrt[4]{2}$D. $\sqrt 2 $
Answer
526.2k+ 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
Master Class 11 Economics: Engaging Questions & Answers for Success

Master Class 11 Accountancy: Engaging Questions & Answers for Success

Master Class 11 English: Engaging Questions & Answers for Success

Master Class 11 Social Science: Engaging Questions & Answers for Success

Master Class 11 Biology: Engaging Questions & Answers for Success

Master Class 11 Physics: Engaging Questions & Answers for Success

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

When and how did Canada eventually gain its independence class 10 social science CBSE

Write a paragraph on any one of the following outlines class 10 english CBSE

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

Which period in Medieval Western Europe is known as class 10 social science CBSE

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